Updated on 2024/04/16

写真a

 
MATSUZAKI, Katsuhiko
 
Affiliation
Faculty of Education and Integrated Arts and Sciences, School of Education
Job title
Professor
Degree
京都大学博士(理学) ( 1992.03 )

Research Experience

  • 2010.04
    -
     

    Waseda University   Faculty of Education and Integrated Arts and Sciences

  • 2006.04
    -
    2010.03

    岡山大学自然科学研究科教授

  • 2005.04
    -
    2006.03

    Ochanomizu University   Faculty of Science

  • 1995.10
    -
    2005.03

    Ochanomizu University   Faculty of Science

  • 1990.10
    -
    1995.09

    Tokyo Institute of Technology   School of Science

Education Background

  •  
    -
    1989

    Kyoto University   Graduate School, Division of Natural Science  

  •  
    -
    1987

    Kyoto University   Faculty of Science  

Professional Memberships

  •  
     
     

    日本数学会

Research Areas

  • Geometry / Basic analysis

Research Interests

  • 複素解析学

  • Hyperbolic Geometry

Awards

  • 解析学賞

    2022.09   日本数学会  

  • 建部賢弘賞

    1996.11   日本数学会  

 

Papers

  • Strongly symmetric homeomorphisms on the real line with uniform continuity

    Huaying Wei, Katsuhiko Matsuzaki

    Indiana University Mathematics Journal   72 ( 4 ) 1553 - 1576  2023.09  [Refereed]

    DOI

  • Parametrization of the p-Weil–Petersson curves: holomorphic dependence

    Huaying Wei, Katsuhiko Matsuzaki

    Journal of Geometric Analysis   33 ( 9 )  2023.09  [Refereed]

     View Summary

    Similar to the Bers simultaneous uniformization, the product of the p-Weil–Petersson Teichmüller spaces for p≥ 1 provides the coordinates for the space of p-Weil–Petersson embeddings γ of the real line R into the complex plane C . We prove the biholomorphic correspondence from this space to the p-Besov space of u= log γ′ on R for p> 1 . From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of p-Weil–Petersson curves. Specifically, it implies that the correspondence of the Riemann mapping parameters to the arc-length parameters preserving the images of curves is a homeomorphism with bi-real-analytic dependence of the change of parameters. This is analogous to the classical theorem of Coifman and Meyer for chord-arc curves.

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    Scopus

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  • The p-integrable Teichmüller space for p≥1

    Huaying Wei, Katsuhiko Matsuzaki

    Proceedings of the Japan Academy Series A: Mathematical Sciences   99 ( 6 ) 37 - 42  2023.06  [Refereed]

     View Summary

    We verify that the p-integrable Teichmuller space T-p admits the canonical complex Banach manifold structure for any p = 1. Moreover, we characterize a quasisymmetric homeomorphism corresponding to an element of T-p in terms of the p-Besov space for any p > 1.

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  • BMO embeddings, chord-arc curves, and Riemann mapping parametrization

    Huaying Wei, Katsuhiko Matsuzaki

    Advances in Mathematics   417  2023.03  [Refereed]

     View Summary

    We consider the space of chord-arc curves on the plane passing through infinity with their parametrization γ defined on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that γ↦log⁡γ′ is a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve the conjecture of Katznelson, Nag, and Sullivan by showing that this dependence is not continuous.

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  • Space of chord-arc curves and BMO/VMO Teichmüller space

    Katsuhiko Matsuzaki, Huaying Wei

    Annales Fennici Mathematici   48 ( 1 ) 27 - 42  2023  [Refereed]

     View Summary

    This paper focuses on the structure of the subspace Tc of the BMO Teichmüller space Tb corresponding to chord-arc curves, which contains the VMO Teichmüller space Tv. We prove that Tc is not a subgroup with respect to the group structure of Tb, but it is preserved under the inverse operation and the left and the right translations by any element of Tv. Moreover, we show that Tb has a fiber structure induced by Tv, and the complex structure of Tb can be projected down to the quotient space Tv\Tb. Then, we see that Tc consists of fibers of this projection, and its quotient space also has the induced complex structure.

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  • The VMO-Teichmüller space and the variant of Beurling–Ahlfors extension by heat kernel

    Huaying Wei, Katsuhiko Matsuzaki

    Mathematische Zeitschrift   302 ( 3 ) 1739 - 1760  2022.11  [Refereed]

     View Summary

    We give a real-analytic section for the Teichmuller projection onto the VMO-Teichmuller space by using the variant of Beurling-Ahlfors extension by heat kernel introduced by Fefferman et al. (Ann Math 134:65-124, 1991). Based on this result, we prove that the VMO-Teichmuller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichmuller space admits a real-analytic contraction mapping.

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  • The p-Weil–Petersson Teichmüller Space and the Quasiconformal Extension of Curves

    Huaying Wei, Katsuhiko Matsuzaki

    Journal of Geometric Analysis   32 ( 8 ) 213  2022.05  [Refereed]

     View Summary

    We consider the correspondence between the space of p-Weil-Petersson curves gamma on the plane and the p-Besov space of u = log gamma' on the real line for p > 1. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for u on a domain of the p-Besov space to the space of p-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmuller projection from the space of p-integrable Beltrami coefficients to the p-Weil-Petersson Teichmiiller space.

    DOI

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  • Teichmüller spaces of piecewise symmetric homeomorphisms on the unit circle

    Huaying Wei, Katsuhiko Matsuzaki

    Pacific Journal of Mathematics   314 ( 2 ) 495 - 514  2021.10  [Refereed]

     View Summary

    We interpolate a new family of Teichmuller spaces T-#(X) between the universal Teichmuller space T and its little subspace T-0. Each T-#(X) is defined by prescribing a subset X of the unit circle as the exceptional set of the vanishing property for T-0. The inclusion relation of X induces a natural inclusion of T-#(X), and an approximation of T by an increasing sequence of T-#(X) is investigated. In this paper, we discuss the fundamental properties of T-#(X) from the viewpoint of the quasiconformal theory of Teichmuller spaces. We also consider the quotient space of T by T-#(X) as an analog of the asymptotic Teichmuller space.

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  • Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion

    Huaying Wei, Katsuhiko Matsuzaki

    Analysis and Mathematical Physics   11 ( 2 )  2021.06  [Refereed]

     View Summary

    A quasisymmetric homeomorphism defines an element of the universal Teichmüller space and a symmetric one belongs to its little subspace. We show an example of a symmetric homeomorphism h of the real line R onto itself such that h- 1 is not symmetric. This implies that the set of all symmetric self-homeomorphisms of R does not constitute a group under the composition. We also consider the same problem for a strongly symmetric self-homeomorphism of R which is defined by a certain concept of harmonic analysis. These results reveal the difference of the sets of such self-homeomorphisms of the real line from those of the unit circle.

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  • Beurling–Ahlfors extension by heat kernel, A∞-weights for VMO, and vanishing Carleson measures

    Huaying Wei, Katsuhiko Matsuzaki

    Bulletin of the London Mathematical Society   53 ( 3 ) 723 - 739  2021.06  [Refereed]

     View Summary

    We investigate a variant of the Beurling–Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (that is, its derivative is an (Formula presented.) -weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half-plane.

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  • Teichmüller spaces of generalized symmetric homeomorphisms

    Huaying Wei, Katsuhiko Matsuzaki

    Proceedings of the American Mathematical Society, Series B   7   52 - 66  2020.05  [Refereed]

     View Summary

    We introduce the concept of a new kind of symmetric homeomorphism on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we show that the Bers Schwarzian derivative map is a holomorphic split submersion and endow a complex Banach manifold structure on the Teichm¨uller space of those generalized symmetric homeomorphisms.

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  • Rigidity of groups of circle diffeomorphisms and teichmüller spaces

    Katsuhiko Matsuzaki

    Journal d'Analyse Mathematique   140 ( 2 ) 511 - 548  2020.03  [Refereed]

     View Summary

    We consider deformations of a group of circle diffeomorphisms with Hölder continuous derivative in the framework of quasiconformal Teichmüller theory and showcertain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a Möbius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichmüller space with the Weil–Petersson metric.

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  • Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

    Katsuhiko Matsuzaki

    Revista Matematica Iberoamericana   36 ( 5 ) 1333 - 1374  2020.02  [Refereed]

     View Summary

    Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with α-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C1+α-topology at the base point.

    DOI

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  • Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space

    Katsuhiko Matsuzaki, Yasuhiro Yabuki, Johannes Jaerisch

    Groups, Geometry, and Dynamics   14 ( 2 ) 369 - 411  2020  [Refereed]

     View Summary

    For a non-elementary discrete isometry group G of divergence type acting on a proper geodesic ı-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of G. As applications of this result, we have: (1) under a minor assumption, such a discrete group G admits no proper conjugation, that is, if the conjugate of G is contained in G, then it coincides with G; (2) the critical exponent of any non-elementary normal subgroup of G is strictly greater than half of that for G.

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  • Weighted cogrowth formula for free groups

    Johannes Jaerisch, Katsuhiko Matsuzaki

    Groups, Geometry, and Dynamics   14 ( 2 ) 349 - 368  2020  [Refereed]

     View Summary

    We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group Cay.Fn/ by an arbitrary subgroup G of Fn. Our main result, which generalizes Grigorchuk’s cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on Gn Cay.Fn/ to the Poincaré exponent of G. Our main tool is the Patterson–Sullivan theory for metric trees.

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  • On horospheric limit sets of Kleinian groups

    Kurt Falk, Katsuhiko Matsuzaki

    Journal of Fractal Geometry   7 ( 4 ) 329 - 350  2020  [Refereed]

     View Summary

    In this paper we partially answer a question of P. Tukia about the size of the difference between the big horospheric limit set and the horospheric limit set of a Kleinian group. We mainly investigate the case of normal subgroups of Kleinian groups of divergence type and show that this difference is of zero conformal measure by using another result obtained here: the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.

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  • Injectivity of the quotient Bers embedding of Teichmüller spaces

    Katsuhiko Matsuzaki

    Annales Academiae Scientiarum Fennicae Mathematica   44 ( 2 ) 657 - 679  2019  [Refereed]

     View Summary

    The Bers embedding of the Teichmüller space is a homeomorphism into the Banach space of certain holomorphic automorphic forms. For a subspace of the universal Teichmüller space and its corresponding Banach subspace, we consider whether the Bers embedding can project down between their quotient spaces. If this is the case, it is called the quotient Bers embedding. Injectivity of the quotient Bers embedding is the main problem in this paper. Alternatively, we can describe this situation as the universal Teichmüller space having an affine foliated structure induced by this subspace. We give several examples of subspaces for which the injectivity holds true, including the Teichmüller space of circle diffeomorphisms with Hölder continuous derivative. As an application, the regularity of conjugation between representations of a Fuchsian group into the group of circle diffeomorphisms is investigated.

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  • Dynamics of teichmüller modular groups and topology of moduli spaces of Riemann surfaces of infinite type

    Katsuhiko Matsuzaki

    Groups, Geometry, and Dynamics   12 ( 1 ) 1 - 64  2018  [Refereed]

     View Summary

    We investigate the dynamics of the Teichmüller modular group on the Teichmüller space of a Riemann surface of infinite topological type. Since the modular group does not necessarily act discontinuously, the quotient space cannot inherit a rich geometric structure from the Teichmüller space. However, we introduce the set of points where the action of the Teichmüller modular group is stable, and we prove that this region of stability is generic in the Teichmüller space. By taking the quotient and completion with respect to the Teichmüller distance, we obtain a geometric object that we regard as an appropriate moduli space of the quasiconformally equivalent complex structures admitted on a topologically infinite Riemann surface.

    DOI

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  • Continuity of the barycentric extension of circle diffeomorphisms with Hölder continuous derivative

    Katsuhiko Matsuzaki

    Transactions of the London Mathematical Society   4 ( 1 ) 129 - 147  2017.12  [Refereed]

     View Summary

    The barycentric extension due to Douady and Earle yields a conformally natural extension of a quasisymmetric self-homeomorphism of the unit circle to a quasiconformal self-homeomorphism of the unit disk. We consider such extensions for circle diffeomorphisms with Hölder continuous derivative and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.

    DOI

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  • Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

    Katsuhiko Matsuzaki, José M. Rodríguez

    Mathematische Nachrichten   290 ( 7 ) 1097 - 1112  2017.05  [Refereed]

     View Summary

    We consider a planar Riemann surface R made of a non-compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in R caused by these points to be uniformly distributed in R in terms of Euclidean geometry. Then we associate a graph G with R by taking the Voronoi diagram for the uniformly distributed cusps and show that G represents certain geometric and analytic properties of R.

    DOI

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  • The hyperbolic metric on the complement of the integer lattice points in the plane

    Katsuhiko Matsuzaki

    New Trends in Aanlysis and Interdisciplinanr Applications     247 - 252  2017  [Refereed]

    DOI

  • The teichmüller space of group invariant symmetric structures on the circle

    Katsuhiko Matsuzaki

    Annales Academiae Scientiarum Fennicae Mathematica   42 ( 2 ) 535 - 550  2017  [Refereed]

     View Summary

    We introduce the quasisymmetric deformation space of a Fuchsian group Γ within the group of symmetric self-homeomorphisms of the circle, and define this as the Teichmüller space AT (Γ) of Γ-invariant symmetric structures. This is another generalization of the asymptotic Teichmüller space, and we verify the basic properties of this space. In particular, we show that AT (Γ) is infinite dimensional, and in fact non-separable if Γ admits a non-trivial deformation, even for a cofinite Fuchsian group Γ.

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  • The Chabauty and the Thurston topologies on the hyperspace of closed subsets

    Katsuhiko Matsuzaki

    Journal of the Mathematical Society of Japan   69 ( 1 ) 263 - 292  2017  [Refereed]

     View Summary

    For a regularly locally compact topological space X of T0 separation axiom but not necessarily Hausdorff, we consider a map σ from X to the hyperspace C(X) of all closed subsets of X by taking the closure of each point of X. By providing the Thurston topology for C(X), we see that σ is a topological embedding, and by taking the closure of σ(X) with respect to the Chabauty topology, we have the Hausdorff compactification X̂ of X. In this paper, we investigate properties of X̂ and C(X̂) equipped with different topologies. In particular, we consider a condition under which a self-homeomorphism of a closed subspace of C(X) with respect to the Chabauty topology is a self-homeomorphism in the Thurston topology.

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  • Asymptotic conformality of the barycentric extension of quasiconformal maps

    Katsuhiko Matsuzaki, Masahiro Yanagishita

    Filomat   31 ( 1 ) 85 - 90  2017  [Refereed]

     View Summary

    We first remark that the complex dilatation of a quasiconformal homeomorphism of a hyperbolic Riemann surface R obtained by the barycentric extension due to Douady-Earle vanishes at any cusp of R. Then we give a new proof, without using the Bers embedding, of a fact that the quasiconformal homeomorphism obtained by the barycentric extension from an integrable Beltrami coefficient on R is asymptotically conformal if R satisfies a certain geometric condition.

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  • Growth and cogrowth of normal subgroups of a free group

    Johannes Jaerisch, Katsuhiko Matsuzaki

    Proceedings of the American Mathematical Society   145 ( 10 ) 4141 - 4149  2017  [Refereed]

     View Summary

    We give a sufficient condition for a sequence of normal subgroups of a free group to have the property that both their growths tend to the upper bound and their cogrowths tend to the lower bound. The condition is represented by planarity of the quotient graphs of the tree.

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  • Uniform convexity, normal structure and the fixed point property of metric spaces

    Katsuhiko Matsuzaki

    Topology and its Applications   196   684 - 695  2015.12  [Refereed]

     View Summary

    We say that a complete metric space X has the fixed point property if every group of isometric automorphisms of X with a bounded orbit has a fixed point in X. We prove that if X is uniformly convex then the family of admissible subsets of X possesses uniformly normal structure and if so then it has the fixed point property. We also show that from other weaker assumptions than uniform convexity, the fixed point property follows. Our formulation of uniform convexity and its generalization can be applied not only to geodesic metric spaces.

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  • The critical exponent, the hausdorff dimension of the limit set and the convex core entropy of a Kleinian group

    Kurt Falk, Katsuhiko Matsuzaki

    Conformal Geometry and Dynamics   19 ( 8 ) 159 - 196  2015  [Refereed]

     View Summary

    In this paper we study the relationship between three numerical invariants associated to a Kleinian group, namely the critical exponent, the Hausdorff dimension of the limit set and the convex core entropy, which coincides with the upper box-counting dimension of the limit set. The Hausdorff dimension of the limit set is naturally bounded below by the critical exponent and above by the convex core entropy. We investigate when these inequalities become strict and when they are equalities.

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  • Certain integrability of quasisymmetric automorphisms of the circle

    Katsuhiko Matsuzaki

    Computational Methods and Function Theory   14 ( 2-3 ) 487 - 503  2014.10  [Refereed]

     View Summary

    Using the correspondence between the quasisymmetric quotient and the variation of the cross-ratio for a quasisymmetric automorphism (Formula presented.) of the unit circle, we establish a certain integrability of the complex dilatation of a quasiconformal extension of (Formula presented.) to the unit disk if the Liouville cocycle for (Formula presented.) is integrable. Moreover, under this assumption, we verify regularity properties of (Formula presented.) such as being bi-Lipschitz and symmetric.

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  • Infinite-dimensional Teichmuller spaces and modular groups

    Katsuhiko Matsuzaki

    Handbook of Teichmuller Theory, Vol Iv    2014

  • An estimate of the maximal dilatations of quasiconformal automorphisms of annuli

    Katsuhiko Matsuzaki

    Complex Variables and Elliptic Equations   58 ( 7 ) 923 - 932  2013.07  [Refereed]

     View Summary

    We introduce a certain extremal problem for quasiconformal automorphisms of annuli and give upper and lower estimates for the minimal value of their maximal dilatations. © 2013 Copyright Taylor and Francis Group, LLC.

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  • Non-divergent infinitely discrete Teichmüller modular transformation

    E. Fujikawa, K. Matsuzaki

    Topics in finite or infinite dimensional complex analysis, Tohoku Univ. Press     97 - 102  2013  [Refereed]

  • No proper conjugation for quasiconvex cocompact groups of Gromov hyperbolic spaces

    Katsuhiko Matsuzaki, Yasuhiro Yabuki

    IN THE TRADITION OF AHLFORS-BERS, VI   590   125 - 136  2013  [Refereed]

     View Summary

    We prove that, if a quasiconvex cocompact subgroup of the isometry group of a Gromov hyperbolic space has a conjugation into itself, then it is onto itself.

    DOI

  • The nielsen realization problem for asymptotic teichm̈uller modular groups

    Ege Fujikawa, Katsuhiko Matsuzaki

    Transactions of the American Mathematical Society   365 ( 6 ) 3309 - 3327  2013  [Refereed]

     View Summary

    Under a certain geometric assumption on a hyperbolic Riemann surface, we prove an asymptotic version of the fixed point theorem for the Teichm̈uller modular group, which asserts that every finite subgroup of the asymptotic Teichm̈uller modular group has a common fixed point in the asymptotic Teichm̈uller space. For its proof, we use a topological characterization of the asymptotically trivial mapping class group, which has been obtained in the authors' previous paper, but a simpler argument is given here. As a consequence, every finite subgroup of the asymptotic Teichm̈uller modular group is realized as a group of quasiconformal automorphisms modulo coincidence near infinity. Furthermore, every finite subgroup of a certain geometric automorphism group of the asymptotic Teichm̈uller space is realized as an automorphism group of the Royden boundary of the Riemann surface. These results can be regarded as asymptotic versions of the Nielsen realization theorem. © 2013 American Mathematical Society.

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  • Large and small covers of a hyperbolic manifold

    Petra Bonfert-Taylor, Katsuhiko Matsuzaki, Edward C. Taylor

    Journal of Geometric Analysis   22 ( 2 ) 455 - 470  2012.04  [Refereed]

     View Summary

    The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of "small" regular covers is sharp but is not attained. We also consider "large" (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence.We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two. © Mathematica Josephina, Inc. 2010.

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  • The Petersson series vanishes at infinity

    Katsuhiko Matsuzaki

    QUASICONFORMAL MAPPINGS, RIEMANN SURFACES, AND TEICHMULLER SPACES   575   299 - 311  2012  [Refereed]

     View Summary

    The Petersson series with respect to a simple closed geodesic c on a hyperbolic Riemann surface R is the relative Poincare series of the canonical holomorphic quadratic differential on the annular cover of R and it defines a holomorphic quadratic differential phi(c)(z)dz(2) on R. For the hyperbolic metric rho(z)|dz| on R, we give an upper estimate of rho(-2)(z(p))|phi(c)(z(P))| in terms of the hyperbolic length of c and the distance of p E R from c.

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  • Stable quasiconformal mapping class groups and asymptotic teichmü ller spaces

    Ege Fujikawa, Katsuhiko Matsuzaki

    American Journal of Mathematics   133 ( 3 ) 637 - 675  2011.06  [Refereed]

     View Summary

    The stable quasiconformal mapping class group is a group of quasiconformal mapping classes of a Riemann surface that are homotopic to the identity outside some topologically finite subsurface. Its analytic counterpart is a group of mapping classes that act on the asymptotic Teichm üller space trivially. We prove that the stable quasiconformal mapping class group is coincident with the asymptotically trivial mapping class group for every Riemann surface satisfying a certain geometric condition. Consequently, the intermediate Teichmüller space, which is the quotient space of the Teichmüller space by the asymptotically trivial mapping class group, has a complex manifold structure, and its automorphism group is geometrically isomorphic to the asymptotic Teichmüllermodular group. The proof utilizes a condition for an asymptotic Teichmüller modular transformation to be of finite order, and this is given by the consideration of hyperbolic geometry of topologically infinite surfaces and its deformation under quasiconformal homeomorphisms. Also these arguments enable us to show that every asymptotic Teichmüller modular transformation of finite order has a fixed point on the asymptotic Teichmüller space, which can be regarded as an asymptotic version of the Nielsen theorem. © 2011 by The Johns Hopkins University Press.

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  • Polycyclic quasiconformal mapping class subgroups

    Katsuhiko Matsuzaki

    Pacific Journal of Mathematics   251 ( 2 ) 361 - 374  2011  [Refereed]

     View Summary

    For a subgroup of the quasiconformal mapping class group of a Riemann surface in general, we give an algebraic condition which guarantees its discreteness in the compact-open topology. Then we apply this result to its action on the Teichmüller space. © 2011 by Pacific Journal of Mathematics.

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  • Twists and Gromov hyperbolicity of riemann surfaces

    Katsuhiko Matsuzaki, José M. Rodríguez

    Acta Mathematica Sinica, English Series   27 ( 1 ) 29 - 44  2011.01  [Refereed]

     View Summary

    The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general. © 2011 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

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  • Checking atomicity of conformal ending measures for kleinian groups

    Kurt Falk, Katsuhiko Matsuzaki, Bernd O. Stratmann

    Conformal Geometry and Dynamics   3 ( 8 ) 116 - 150  2010.06  [Refereed]

     View Summary

    In this paper we address questions of continuity and atomicity of conformal ending measures for arbitrary non-elementary Kleinian groups. We give sufficient conditions under which such ending measures are purely atomic. Moreover, we will show that if a conformal ending measure has an atom which is contained in the big horospherical limit set, then this atom has to be a parabolic fixed point. Also, we give detailed discussions of nontrivial examples for purely atomic as well as for non-atomic conformal ending measures. © 1999 American Mathematical Society.

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  • The action of elliptic modular transformations on asymptotic Teichmüller spaces

    Katsuhiko Matsuzaki

    Teichmüller Theory and Moduli Problem, Ramanujan Math. Soc. Lecture Notes Series 10     481 - 488  2010  [Refereed]  [Invited]

  • Symmetric groups that are not the symmetric conjugates of Fuchsian groups

    Katsuhiko Matsuzaki

    IN THE TRADITION OF AHLFORS-BERS, V   510   239 - 247  2010  [Refereed]

     View Summary

    A symmetric automorphism of the unit circle is the boundary extension of an asymptotically conformal automorphism of the unit disk. A symmetric group is a quasisymmetric group whose elements are symmetric automorphisms. In this paper, we consider a problem whether a symmetric group is conjugate to a Fuchsian group by a symmetric homeomorphism or not. Our answer is negative.

  • The Patterson-Sullivan measure and proper conjugation for Kleinian groups of divergence type

    Katsuhiko Matsuzaki, Yasuhiro Yabuki

    Ergodic Theory and Dynamical Systems   29 ( 2 ) 657 - 665  2009.04  [Refereed]

     View Summary

    A Kleinian group (a discrete subgroup of conformal automorphisms of the unit ball) G is said to have proper conjugation if it contains the conjugate αGα-1 by some conformal automorphism α as a proper subgroup in it. We show that a Kleinian group of divergence type cannot have proper conjugation. Uniqueness of the PattersonSullivan measure for such a Kleinian group is crucial to our proof. © 2008 Cambridge University Press.

    DOI

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  • Dynamics on teichmüller spaces and self-covering of riemann surfaces

    Ege Fujikawa, Katsuhiko Matsuzaki, Masahiko Taniguchi

    Mathematische Zeitschrift   260 ( 4 ) 865 - 888  2008.12  [Refereed]

     View Summary

    A non-injective holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmüller space. We investigate the dynamics of such self-embeddings by applying our structure theorem of self-covering of Riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the Teichmüller space. We also extend our observation to quasiregular self-covers of Riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self-cover. © 2008 Springer-Verlag.

    DOI

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    1
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  • Invariance of the Nayatani metrics for Kleinian manifolds

    Katsuhiko Matsuzaki, Yasuhiro Yabuki

    Geometriae Dedicata   135 ( 1 ) 147 - 155  2008.08  [Refereed]

     View Summary

    The Nayatani metric g N is a Riemannian metric on a Kleinian manifold M which is compatible with the standard flat conformal structure. It is known that, for M corresponding to a geometrically finite Kleinian group, g N has large symmetry: the isometry group of (M, g N ) coincides with the conformal transformation group of M. In this paper, we prove that this holds for a larger class of M. In particular, this class contains such M that correspond to Kleinian groups of divergence type. © 2008 Springer Science+Business Media B.V.

    DOI

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    2
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    (Scopus)
  • Elliptic quasiconformal mapping classes acting on asymptotic Teichmüller spaces

    E. Fujikawa, K. Matsuzaki

    Complex Analysis and its Applications, Proceedings of the 15th ICFIDCAA at Osaka City University, OCAMI Studies 2     169 - 173  2008  [Refereed]

  • On quasiconformal invariance of convergence and divergence types for Fuchsian groups

    Katsuhiko Matsuzaki

    Illinois Journal of Mathematics   52 ( 4 ) 1249 - 1258  2008  [Refereed]

     View Summary

    We characterize convergence and divergence types for Fuchsian groups in terms of the critical exponent of convergence and modified functions of the Poincaré series for certain subgroups associated with ends of the quotient Riemann surfaces. As an application of this result, we prove that convergence and divergence type are not invariant under a quasiconformal automorphism of the unit disk. © 2009 University of Illinois.

    DOI

    Scopus

  • Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces

    Katsuhiko Matsuzaki

    Journal d'Analyse Mathematique   102 ( 1 ) 1 - 28  2007.08  [Refereed]

     View Summary

    For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R. © 2007 The Hebrew University of Jerusalem.

    DOI

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    6
    Citation
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  • A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space

    Katsuhiko Matsuzaki

    Proceedings of the American Mathematical Society   135 ( 8 ) 2573 - 2579  2007.08  [Refereed]

     View Summary

    For an analytically infinite Riemann surface R, the quasiconformal mapping class group MCG(R) always acts faithfully on the ordinary Teichmüller space T(R). However in this paper, an example of R is constructed for which MCG(R) acts trivially on its asymptotic Teichmüller space AT (R). © 2007 American Mathematical Society.

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    4
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    (Scopus)
  • Non-stationary and discontinuous quasiconformal mapping class groups

    Ege Fujikawa, Katsuhiko Matsuzaki

    Osaka Journal of Mathematics   44 ( 1 ) 173 - 185  2007.03  [Refereed]

     View Summary

    Every stationary subgroup of the quasiconformal mapping class group of a Riemann surface acts on the Teichmüller space discontinuously if the surface satisfies a certain geometric condition. In this paper, we construct such a Riemann surface that the quasiconformal mapping class group is non-stationary but it still acts on the Teichmüller space discontinuously.

  • The interior of discrete projective structures in the Bers fiber

    Katsuhiko Matsuzaki

    Annales Academiae Scientiarum Fennicae Mathematica   32 ( 1 ) 3 - 12  2007  [Refereed]

     View Summary

    The space of all projective structures on a closed surface is a holomorphic vector bundle over the Teichmüller space. In this paper, we restrict the space to the Bers fiber over any fixed underlying complex structure and prove that the interior of the set of discrete projective structures in the Bers fiber consists of those having quasifuchsian holonomy.

  • A classification of the modular transformations of infinite dimensional Teichmuller spaces

    Katsuhiko Matsuzaki

    In the Tradition of Ahlfors-Bers, IV   432   167 - 177  2007  [Refereed]

     View Summary

    We classify the modular transformations of infinite dimensional Teichmuller spaces according to the behavior of their orbits. We then consider two classes, stationary and asymptotically elliptic, whose elements have a certain property similar to that of the modular transformations of finite dimensional Teichmuller spaces.

  • Recurrent and periodic points for isometries of L spaces

    Ege Fujikawa, Katsuhiko Matsuzaki

    Indiana University Mathematics Journal   55 ( 3 ) 975 - 997  2006  [Refereed]

     View Summary

    We study the action of isometries on metric spaces. In particular, we consider the recurrent set of the bilateral shift operator on the Banach space L∞ (ℤ), and prove that the set of periodic points is not dense in the recurrent set. Then we apply this result to investigating the dynamics of Teichmüller modular groups acting on infinite dimensional Teichmüller spaces as well as composition operators acting on Hardy spaces. Indiana University Mathematics Journal ©.

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    4
    Citation
    (Scopus)
  • A countable Teichmüller modular group

    Katsuhiko Matsuzaki

    Transactions of the American Mathematical Society   357 ( 8 ) 3119 - 3131  2005.08  [Refereed]

     View Summary

    We construct an example of a Riemann surface of infinite topological type for which the Teichmüller modular group consists of only a countable number of elements. We also consider distinguished properties which the Teichmüller space of this Riemann surface possesses. ©2004 American Mathematical Society.

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    14
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  • Isoperimetric constants for conservative fuchsian groups

    Katsuhiko Matsuzaki

    Kodai Mathematical Journal   28 ( 2 ) 292 - 300  2005  [Refereed]

     View Summary

    The critical exponents of conservative Fuchsian groups are bounded from below by 1/2. It is proved in this note that this result is sharp by giving a sequence of conservative Fuchsian groups whose critical exponents converge to 1/2. The proof is carried out by estimating the isoperimetric constants of hyperbolic surfaces associated with the Fuchsian groups. © 2005, Department of Mathematics, Tokyo Institute of Technology. All rights reserved.

    DOI

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    9
    Citation
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  • Indecomposable continua and the limit sets of Kleinian groups

    Katsuhiko Matsuzaki

    In the tradition of Ahlfors and Bers, III, Contemporary Math.   355   321 - 332  2004  [Refereed]

    DOI

  • Inclusion relations between the Bers embeddings of Teichmüller spaces

    Katsuhiko Matsuzaki

    Israel Journal of Mathematics   140   113 - 123  2004  [Refereed]

     View Summary

    We prove that if the Bers embeddings of the Teichmüller spaces of infinitely generated Fuchsian groups are coincident, then these Fuchsian groups are the same.

    DOI

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    13
    Citation
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  • The Infinite Direct Product of Dehn Twists Acting on Infinite Dimensional Teichmuller Spaces

    Katsuhiko Matsuzaki

    Kodai Mathematical Journal   26 ( 3 ) 279 - 287  2003  [Refereed]

    DOI

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    7
    Citation
    (Scopus)
  • Conservative action of Kleinian groups with respect to the Patterson-Sullivan measure

    Katsuhiko Matsuzaki

    Computational Methods and Function Theory   2   469 - 479  2002  [Refereed]

  • Simply connected domains on a hyperbolic surface

    Katsuhiko Matsuzaki

    New Zealand Journal of Mathematics   31   159 - 164  2002  [Refereed]

  • Dynamics of Kleinian groups --- the Hausdorff dimesion of limit sets

    Katsuhiko Matsuzaki

    Selected papers on classical analysis, AMS Translations   204   23 - 44  2001  [Refereed]  [Invited]

  • Convergence of the Hausdorff dimension of the limit sets of Kleinian groups

    Katsuhiko Matsuzaki

    In the tradition of Ahlfors and Bers, Contemporary Math.   256   243 - 254  2000  [Refereed]

  • The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups

    Katsuhiko Matsuzaki

    Mathematical Proceedings of the Cambridge Philosophical Society   128 ( 1 ) 123 - 139  2000.01  [Refereed]

     View Summary

    In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension. © 2000 Cambridge Philosophical Society.

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    7
    Citation
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  • The isomorphism theorem of Kleinian groups

    Katsuhiko Matsuzaki

    Analysis and Topology, World Scientific     507 - 513  1998  [Refereed]

  • Structural stability of Kleinian groups

    K Matsuzaki

    MICHIGAN MATHEMATICAL JOURNAL   44 ( 1 ) 21 - 36  1997  [Refereed]

    DOI

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    1
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  • The Petersson series for short geodesics

    Katsuhiko Matsuzaki

    Proceedings of the XVI Rolf Nevanlinna Colloquium, Walter de Gruyter     143 - 150  1996  [Refereed]

  • Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces

    Katsuhiko Matsuzaki

    Geometric Complex Analysis, World Scientific     443 - 450  1996  [Refereed]  [Invited]

    CiNii

  • Conformal conjugation of Fuchsian groups from the first kind to the second kind

    K Matsuzaki, H Shiga

    JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK   476 ( 476 ) 191 - 200  1996  [Refereed]

  • TEICHMULLER-SPACES WITH VARIABLE BASES IN THE UNIVERSAL TEICHMULLER SPACE

    K MATSUZAKI

    ANNALES ACADEMIAE SCIENTIARUM FENNICAE SERIES A1-MATHEMATICA   20 ( 1 ) 27 - 36  1995  [Refereed]

     View Summary

    It is proved that the embeddings of Teichmuller spaces of cofinite Fuchsian groups of distinct moduli are discrete in the universal Teichmuller space.

  • The conservative-dissipative dichotomy for geometric covers of Riemann surfaces

    Katsuhiko Matsuzaki

    Revue Roumaine de Mathématiques Pures et Appliquées   40   77 - 80  1995  [Refereed]

  • Projective structures inducing covering maps

    Katsuhiko Matsuzaki

    Duke Mathematical Journal   78 ( 2 ) 413 - 425  1995  [Refereed]

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • NOTES ON PROJECTIVE-STRUCTURES AND KLEINIAN-GROUPS

    K MATSUZAKI, JA VELLING

    OSAKA JOURNAL OF MATHEMATICS   31 ( 1 ) 165 - 175  1994.03  [Refereed]

  • ERGODIC PROPERTIES OF DISCRETE-GROUPS - INHERITANCE TO NORMAL-SUBGROUPS AND INVARIANCE UNDER QUASI-CONFORMAL DEFORMATIONS

    K MATSUZAKI

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   33 ( 1 ) 205 - 226  1993.02  [Refereed]

  • Simply connected invariant domains of Kleinian groups not in the closures of Teichmüller spaces

    Katsuhiko Matsuzaki

    Complex Variables   22   93 - 100  1993  [Refereed]

    CiNii

  • THE ACTION AT INFINITY OF CONSERVATIVE GROUPS OF HYPERBOLIC MOTIONS NEED NOT HAVE ATOMS

    JA VELLING, K MATSUZAKI

    ERGODIC THEORY AND DYNAMICAL SYSTEMS   11 ( 3 ) 577 - 582  1991.09  [Refereed]

     View Summary

    Herein the authors show that discrete groups of motions on Hn+1 may be conservative on S(n) but have no positive measure ergodic components for this boundary action. An explicit example of such a group is given for H-3 using the Apollonian circle packing of R2.

    DOI

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    2
    Citation
    (Scopus)
  • A CHARACTERIZATION OF EXTENDED SCHOTTKY TYPE-GROUPS WITH A REMARK TO AHLFORS CONJECTURE

    K MATSUZAKI

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   31 ( 1 ) 259 - 264  1991.02  [Refereed]

  • Geometric finiteness, quasiconformal stability and surjectivity of the bers map for kleinian groups

    Katsuhiko Matsuzaki

    Tohoku Mathematical Journal   43 ( 3 ) 327 - 336  1991  [Refereed]

    DOI

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    2
    Citation
    (Scopus)
  • Certain estimates on kleinian groups by the core of their quotient 3-manifold

    Katsuhiko Matsuzaki

    Kodai Mathematical Journal   13 ( 3 ) 377 - 385  1990  [Refereed]

    DOI

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▼display all

Books and Other Publications

  • Topics in finite or infinite dimensional complex analysis : proceedings of the 19th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, December 11-15, 2011, Aster Plaza, Hiroshima, Japan

    Katsuhiko Matsuzaki, Toshiyuki Sugawa

    Tohoku University Press  2013 ISBN: 9784861632198

  • Hyperbolic Manifolds and Kleinian Groups

    Katsuhiko Matsuzaki, Masahiko Taniguchi

    Oxford University Press  1998.04 ISBN: 9780198500629

  • 双曲的多様体とクライン群

    谷口 雅彦, 松崎 克彦

    日本評論社  1993 ISBN: 9784535782020

Research Projects

  • レブナー方程式とタイヒミュラー空間論

    日本学術振興会  科学研究費助成事業 基盤研究(B)

    Project Year :

    2023.04
    -
    2028.03
     

    松崎 克彦

  • 画像処理における2次元曲線の変形の効率化と等角接合による認証

    日本学術振興会  科学研究費助成事業

    Project Year :

    2023.06
    -
    2026.03
     

    松崎 克彦

  • Quasiconformal extension in differential geometry and theory of the universal Teichmueller space in harmonic analysis

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2018.04
    -
    2023.03
     

  • Theory of the universal Teichmüller space in harmonic analysis

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for JSPS Fellows

    Project Year :

    2021.04
    -
    2023.03
     

  • Analysis of elliptic operators and its applications to Geometric Function Theory

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2017.04
    -
    2022.03
     

  • Researches on the spectrum of critical exponents of normal subgroups and the rigidity of cogrowth for hyperbolic discrete groups

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research

    Project Year :

    2016.04
    -
    2020.03
     

    Matsuzaki Katsuhiko

     View Summary

    The Grigorchuk cogrowth formula gave a relationship between the exponent of convergence for the isometric action of normal subgroups on the Cayley graph of the free group and the bottom of the spectrum of the discrete Laplacian on the quotient graph. Similar results were proved by Sullivan for Kleinian groups acting on the hyperbolic space and for the Laplacian on the hyperbolic manifold. A common phenomenon is that the phase transition of the exponents occurs at 1/2 of that of the base group. In this study, even when the lengths of edges of the Cayley graph of a free group is varied, a weighted discrete Laplacian depending on the exponent of convergence of a normal subgroup determines the spectrum, and the generalization of the cogrowth formula was proved for this. The phase transition at 1/2 of the exponent was also verified in this case.

  • Research on the Weil-Petersson metric of infinite dimensional Teichmueller spaces

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2013.04
    -
    2019.03
     

    Matsuzaki Katsuhiko, YANAGISHITA Masahiro

     View Summary

    We introduced the Teichmueller space of diffeomorphisms of the unit circle with Hoelder continuous derivatives as a subspace of the universal Teichmueller space. We provided a complex Banach manifold structure for it and proved that its topology coincides with the one induced by the Hoelder constants of the maps. We also proved that the barycentric extension induces a continuous section from the Teichmueller space of the circle diffeomorphisms with respect to this topology. Then, we considered deformations of a group of circle diffeomorphisms with Hoelder continuous derivative and showed certain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a Moebius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichmueller space with the Weil-Petersson metric.

  • Research on the Teichmuller spaces of fractal structures

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2015.04
    -
    2018.03
     

    Taniguchi Masahiko, Fujimura MASAYO, Matsuzaki KATSUHIKO, Fujikawa EGE

     View Summary

    We formulate the concept of the Teichmuller space of a fractal structure and establish the fundamental theory on it. This is one of the main purposes of this research project. More precisely, we introduce the Teichmuller space of a countable set of points associated with the fractal structure on a general Riemann surface. Furthermore, we show that such a space admits a natural complex analytic structure if the fractal structure possesses standard bounded geometry.
    The second purpose of this research project is to introduce geometric global coordinates for such a Teichmuller space. On this point, for several important cases such as the iterated function systems by Mobius transformations, Kleinian group actions, and infinitely generated Koebe group actions, we introduce natural geometric global coordinates on the Teichmuller space of the corresponding fractal structure, and obtain a global representation of it.

  • Thermodynamic formalism for conformal semigroup actions

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Research Activity Start-up

    Project Year :

    2015.08
    -
    2017.03
     

    Jaerisch Johannes, SUMI Hiroki, MATSUZAKI Katsuhiko, KESSEBÖHMER Marc, STADLBAUER Manuel, MUNDAY Sara

     View Summary

    We have investigated the interplay of dynamics and geometry of conformal semigroup actions. Two of the main results are the following.
    1) We have applied the multifractal analysis to random complex dynamical systems. In particular, we have investigated the Hoelder exponent of the long-term behavior depending on the initial value (J. Jaerisch, H. Sumi, Adv. Math 313 (2017), 36 pages).
    2) We have investigated the Poincare exponent of abstract free groups and its normal subgroups. In the proof we use the discrete Laplacian on the associated Cayley graph (J. Jaerisch, K. Matsuzaki, Proc. AMS, to appear 2017).

  • Resolution of conjugation problems on circle diffeomorphism groups

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research

    Project Year :

    2012.04
    -
    2016.03
     

    Matsuzaki Katsuhiko, TANIGUCHI MASAHIKO, FUJIKAWA EGE

     View Summary

    (1) We introduced the Teichmueller space of circle diffeomorphisms with Hoelder continuous derivatives and established its foundation. (2) We proved that if a group of Moebius transformations is conjugate to a group of circle diffeomorphisms with Hoelder continuous derivatives by a symmetric homeomorphism, then the conjugating map actually has a Hoelder continuous derivative of the same order. (3) We obtained a necessary and sufficient condition for a group of circle diffeomorphisms with α-Hoelder continuous derivatives for α>1/2 to be conjugate to a group of Moebius transformations by a circle diffeomorphism with an α-Hoelder continuous derivative in terms of uniform integrability of the complex dilatations of quasiconformal extensions of the group elements. (4) Even if we do not assume α>1/2, we showed that if the integrability of quasiconformal extensions is uniformly bounded by a certain constant sufficiently small, then the above result still holds true.

  • 熱力学形式によるクライン群の幾何の研究

    日本学術振興会  科学研究費助成事業 特別研究員奨励費

    Project Year :

    2014.04
    -
    2015.03
     

    松崎 克彦, JAERISCH Johannes

     View Summary

    はじめに,フックス群および自由群の非自明正規部分群で,収束指数がもとの群の 1/2 となるものの例の構成を試みた.収束指数を商空間のラプラシアンのスペクトルの底で読み替えて,それを幾何学的に評価する方針をとったが,等周定数を用いる方法では原理的に不可能であることがわかった.ラプラシアンの固有関数を構成して,スペクトルの底を直接に評価することも成功しなかった.収束指数が最大指数の 1/2 以下となる群の構成法がほとんど知られていないこと,およびある軌道に関する反転で生成される群の収束指数をもとの軌道に関する収束指数で評価する問題が重要であることが判明した.
    <br>
    その後,収束指数がもとの群の 1/2 に近づく非自明正規部分群の列の構成を自由群の場合に考察した.方法はやはりスペクトルの底を等周定数を用いて評価するのであるが, Mohar によるグラフ理論の結果で,スペクトルの底は等周定数を用いて評価できることがわかった.さらに,等周定数は平面グラフの場合には単射半径で評価できることを示した.結論としては,自由群の生成元の十分大きなべきで生成される正規部分群の列をとれば,収束指数がもとの群の 1/2 に近づくことが証明できた.
    <br>
    (相対)双曲群の非自明正規部分群による剰余類群の増大度(収束指数)に関するW. Yang の結果に,これがもとの双曲群の収束指数に近づくような群の列を構成するものがある.剰余類群の増大度と,上で述べた商空間のラプラシアンのスペクトルの底および等周定数の間の関係は,Mohar による同じ論文で研究されている.これにより,自由群の場合には剰余類群の増大度の問題は,非自明正規部分群の収束指数(双対増大度)に関する研究結果からも従うことがわかった.さらに,自由群の収束指数,非自明正規部分群の双対増大度および剰余類群の増大度の間に成立する関係式を導くことができた.

  • Various aspects on the study of Geometric Function Theory

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2010.04
    -
    2015.03
     

    SUGAWA Toshiyuki, SHIGA Hiroshige, YANAGIHARA Hiroshi, SAKAN Ken-ichi, MATSUZAKI Katsuhiko, FUJIKAWA Ege, MIZUTA Yoshihiro, TANIGUCHI Masahiko, FUJIWARA Koji, ABE Makoto

     View Summary

    Geometric Function Theory deals with problems finding relations between geometrically described (simply-connected) domains in the plane and analytic properties of conformal mappings onto them. Bieberbach conjectured that, for a conformal mapping f(z)=a_0+a_1z+a_2z^2+... normalized by a_0=0, a_1=1, the modulus |a_n| of a_n is not greater than n.This conjecture was finally proved by de Branges in his 1985 paper. In the present research project, for example, we showed that the set of conformal mappings as above with half-integral coefficients consists of exactly 21 functions. The similar set for the integral coefficients was previously known to consist of 9 functions.

  • Teichmuller spaces of symmetric structures and the rigidity and fixed-point problems of quasiconformal mapping class groups

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2008.04
    -
    2012
     

    MATSUZAKI Katsuhiko, TANIGUCHI Masahiko, SUGAWA Toshiyuki, SAKAN Kenichi, SHIGA Hiroshige, NAKANISHI Toshihiro, MIYACHI Hideki, ITO Kentaro, FUJIKAWA Ege

     View Summary

    We define various classes for self-homeomorphisms of the unit circle from a viewpoint of their quasiconformal extension, and study the parameter spaces of those families, which are regarded as Teichmuller spaces. In particular, for the Teichmuller space of symmetric homeomorphisms, we consider conditions for a group acting on this space to have a fixed point in it. As an application, we give a condition for a group of diffeomorphisms to be conjugate to the canonical group action of the circle (Mobius transformations) as well as a condition for the deformation of such a group to be trivial (that is, to have rigidity).

  • 複素力学系の群論への応用:Burnside問題とHopf問題

    岡山大学・早稲田大学  科学研究費 萌芽研究

    Project Year :

    2008.04
    -
    2011.03
     

    松崎克彦

     View Summary

    グロモフ双曲空間に作用する等長変換からなる離散群について,等長変換による共役で与えられる自己単射準同型に関するco-Hopf問題を考えた.co-Hopf問題とは自己単射準同型が全射となる条件をさがす問題である.昨年度以来,quasiconvex cocompact群に対しては,共役で与えられる自己単射準同型が全射となることを示す議論を得ていたが,今年度はその細部を精査し,論文にまとめ,講演として発表することができた.また,古典的双曲空間のクライン群の場合はより広く発散型の群に対して成立するので,グロモフ双曲空間でもそれを目標とした.そのために解決するべき問題は,擬等角不変測度の一意性の適切な定式化にあることが解明できた.
    写像類群の極限集合の孤立点とBurnside問題については,写像類群の固定点集合のある性質を仮定すれば孤立点の存在が証明できるところまではわかった.写像類群は位相的無限型のリーマン面のに対してはタイヒミュラー空間には不連続に作用するとは限らず,極限集合がタイヒミュラー空間内に定義される.多くの場合は極限集合は完全集合となる.しかし,写像類群の部分群としてリーマン面の等角自己同型群を考えると,極限集合が孤立点をもつためには,それは無限群であるがすべての真部分群が有限群であるような群を指数有限に含む必要があることがわかる.このような有限生成群はBurnside問題として研究されperiodic groupとよばれている.楕円モジュラー群の主合同部分群の剰余類群としてperiodic groupを実現すれば,対応するリーマン面の等角自己同型群としてそれはあらわれる.写像類群の非自明な元の固定点集合全体が閉集合であることを仮定すれば,このようなタイヒミュラー空間に対して写像類群の極限集合が孤立点をもつことが証明できた.

  • グロタンディークデッサンと非合同的タイヒミュラー被覆の数論

    科学研究費助成事業(岡山大学)  科学研究費助成事業(萌芽研究)

    Project Year :

    2007
    -
    2009
     

    中村 博昭, 鳥居 猛, 鈴木 武史, 吉野 雄二, 山田 裕史, 松崎 克彦, 廣川 真男, 石川 佳弘

     View Summary

    昨年度に基礎を確立した複素および1進の反復積分の関数等式の導出法(Wojtkowiak氏との共同研究)を延長して,具体的な実例計算をさらに検証した.とりわけ古典的な高次対数関数について知られている分布関係式(distribution relation)の1進版を導出することに成功した.分布関係式は,様々な特殊値を代入することで,高次対数関数の特殊値の間に成立する様々な関係式を組織的に生み出す重要なものであり,1進の場合にも並行してガロア群上の関数族(1-コチェイン)を統御する要となることが期待されるが,前年度までに得られた関数等式との整合性についても検証を行った.8月にケンブリッジのニュートン数理科学研究所で行われた研究集会"Anabelian Geometry"において口頭発表を行った.このときの講演に参加していたH.Gangl氏,P.Deligne氏から今後の研究指針を考える上で有用になると思われるコメントを頂戴することが出来た.また分布関係式の低次項の解消問題に関連して,有理的な道に沿った解析接続の概念にっいて考察を進める必要が生じた.こうしたテーマに関連して研究分担者の鳥居氏には,有理ホモトピー論に関する情報収集を担当していただき,また研究分担者の鈴木氏には,量子代数やKZ方程式との関連で組みひも群の数理についての情報収集を担当していただいた.以上の研究成果の一部は,共同研究者のWojtkowiak氏と協力して,"On distribution formula of complex and 1-adic polylogarithms"という仮題の草稿におおよその骨子をまとめたが,まだ完成に至っていない.周辺にやり残した問題(楕円ポリログ版など)もあり,これらについて一定の目処をつけてから公表までの工程を相談する予定になっている.

  • New development of geometric function theory focused on conformal invariants

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2005
    -
    2008
     

    SUGAWA Toshiyuki, MIZUTA Yoshihiro, SAKAN Ken-ichi, SHIBA Masakazu, YOSHINO Masafumi, TANIGUCHI Masahiko, SHIGA Hiroshige, MATSUZAKI Katsuhiko, NAKAHISNI Toshihiro, SHIMOMURA Tetsu

  • Research on deformations of real and complex manifolds and variations of invariants

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)

    Project Year :

    2005
    -
    2008
     

    SHIGA Hiroshige, AIKAWA Hiroaki, SUGAWA Toshiyuki, MURATA Minoru, MATSUZAKI Katsuhiko, NOGUCHI Junjiro, MIYAJIMA Kimio, IMAYOSHI Yoichi, SHISHIKURA Mitsuhiro, SUMI Hiroki

  • Researches on quasiconformal groups and the modular group of the universal Teichmuller space

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2004.04
    -
    2007
     

    MATSUZAKI Katsuhiko, TANIGUCHI Masahiko, NAKANISHI Toshihiro, SHIGA Hiroshige, SUGAWA Toshiyuki, SAKAN Kenichi

     View Summary

    The Teichmuller Tare is a deformation space of the conformal structures of a Riemann surface. The quasiconformal mapping class group is a certain quotient group of the quasiconformal homeomorphisms of the Riemann surface and it acts on the Teichmuller space as the group of biholomorphic automorphisms (modular transformations). When Teichmuller spaces are finite dimensional, they are widely studied with great importance in various fields of mathirnatics. We aim to extend them to infinite dimensional Teichmuller sperms. In this research, we investigated the dynamics of the quasiconformal mapping class on the Teichmuller space. For this purpose, we also considered a certain quotient space of the Teichmuller space, which is called the asymptotic Teichmuller space.
    We first investigated the recurrent set for the mapping class group and proved that the periodic points are not dense in this set. This result was a foundation of our further studies on the action of elliptic modular transformation (conformal mapping classes) and the classification of the modular transformations. Our classification was based on the behavior of the orbit and we specified two classes, which have a similar nature of the modular transformations of finite dimensional Teichmuller spaces. One is a class of stationary mapping classes, and the other is a class of modular transformations that have a fixed point on the asymptotic Teichmuller space. We noticed that the action of a stationary mapping class group is stable, but also gave an example of a non-stationary mapping class group that acts discontinuously. As an extreme case, we dealt with a mapping class group that has a common fixed point on the asymptotic Teichmuller space and proved that such a group consists of countably many elements.
    As another topic, we studied holomorphic self-covering of Riemann surfaces. We gave a necessary condition for a hyperbolic Riemann surface to admit a (non-injective) holomorphic self-cover in terms of the corresponding Fuchisian group. Namely, if the Fuchsian group is of divergence type at the critical exponent of its Poincare series, then the Riemann surface has no self-covers. The proof used uniqueness of the Patterson-Sullivan measure and can be extended to higher dimensional cases. A holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmuller space. We investigated the dynamics of such a self-embedding and examined the distribution of isometric tangent vectors over Teichmuller space. We also extended our observation to quasiregular self-covers of Riemann surfaces.

  • Dynamics of modular groups on infinite dimensional Teichmuller spaces

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2002.04
    -
    2003
     

    MATSUZAKI Katsuhiko, SUGAWA Toshiyuki

     View Summary

    Teichmueller spaces are not homogeneous spaces and their mudular groups do not act transitively. For compact Riemann surfaces, modular groups act discontinuously, but this is not the case for infinite dimensional Teichmueller spaces. We study the moduli spaces of Riemann surafces of infinite type by considering the chaotic behavior of the action of modular groups. For a viewpoint of general topology, the moduli space is either metrizable or not of the first separation axiom. However, except for a singular part, it can possess a certain geometric structure. In this research, we characterize this stable region by hyperbolic geometric structure of a Riemann surface and construct a contracted moduli space by the completion of the stable region. Consequently, we can describe the closure of a point set in terms of the geomery of Riemann surfaces, which is a point of teh contracted module space.
    We considered the space of pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian model of the universal Teichmueller space and classical families of univalent functions. We also investigated geometric properties of univalent functions with a prescribed growth of the Schwarzian derivative and found that they are starlike or convex according to the distance to the origin in the Bers embedding of the universal Teichmueller space.

  • リーマン面上の射影構造の離散的ホロノミー表現の研究

    お茶の水女子大学  科学研究費 奨励研究(A)

    Project Year :

    2000.04
    -
    2002.03
     

    松崎克彦

     View Summary

    閉曲面S上に入る射影構造、すなわち局所的にリーマン球面をモデルとし,座標変換がメビウス変換であるような幾何構造を考える.リーマン面上の射影構造全体の空間は,その上の正則2次微分全体のなすベクトル空間と同一視することができるが,リーマン面の複素構造も変形して面S上,の全射影構造の空間を考えると,タイヒミュラー空間を底空間とし,各ファイバーが正則2次微分の複素ベクトル空間である解析的バンドルP(S)が得られる.P(S)からSの基本群のPSL(2, C)表現空間への写像で,射影構造に対してそのホロノミー表現を対応させたものをホロノミー写像という.面の基本群の離散表現空間は複素力学系理論における有理関数のマンデルブロー集合に相当するものである.これを擬等角写像等の複素解析的方法と,面上の双曲構造およびPSL(2, C)表現に対応して現れる3次元双曲多様体の幾何学を用いて解析した.
    マンデルブロー集合の境界の解析のためには,擬フックス群の場合に射影構造の構成法の一意性を述べたGoldmanの定理を,ホロノミー表現が全退化群となるものにも拡張することが必要になった.このためには,展開写像から決まるある種の領域がリーマンのを位相的には単純に分割していることを示し,その分割から展開写像の構成法に関する情報を引き出すことが問題であった.極限集合の局所連結性は仮定できないので,古典的な平面上の等角写像の境界挙動の解析を用いた新しい手作りの議論が要求された.Goldmanの定理の拡張のためのプログラムが公表でき,いくつかのステップを設定し,リーマン面上の単連結領域に関する論文を書いた.しかし,全退化群の極限集合を考える過程で,連続体の分解可能性という概念が議論のために本質的であることにはじめて気付いた.これはgeneral topologyにおける問題であったが,それ自身膨大な研究がされている分野であると同時に,力学系の理論でも特異集合の分解可能性が問題の本質になっている場合が多い.実際,複素力学系のジュリア集合の分解可能性についてもRogersによる一連の仕事が既になされていた.それをクライン群の場合に焼き直した論文を書いた.

  • Analysis of Complex Dynamics

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

    Project Year :

    2001
    -
    2002
     

    TANIGUCHI M, SHIGA H, KISAKA M, KOKUBU H, MATSUZAKI K

     View Summary

    The head investigator, M. Taniguchi introduced a new kind of combinatorial model for the covering structure of an entire function, which is called "a configuration tree". This concept is the counterpart of that of the Cayley graph of a Kleinian group in the context of the Sullivan's dictionary.
    This new model enabled us to find a very important class of entire functions, which is called the class of structurally finite entire functions. We also discovered that every element of this class has the very explicit representation as a primitive function of a decorated exponential function. As a consequence, we also proved that the Julia set of every structurally finite entire function has the Hausdorff dimension two.
    Furthermore, very recently we proved the Bell conjecture on explicit representation of multi-connected planar domains affirmatively by using the Hurwitz space, which is a kind of the representation space of holomorphic branched covering structures.

  • Potential theory in a domain with fractal boundary

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1999
    -
    2001
     

    WATANABE Hisako, MAEDA Michie, MATSUZAKI Katsuhiko, TAKEO Fukiko, YOSHIDA Hidenobu

     View Summary

    We consider the boundary-value problems in a domain D with fractal boundary. It often occurs that an operator K on the Besov space on the boundary is bounded with respect to the Besov norms. We can prove the boundedness of an operator from δD to δD in the following method.
    (1) We extend a function defined on δD to R^n by using an extension operator E.
    (2) The Besov norm of f is estimated by (∫_D |▽f(x)|^<Pλ>dx)^<1/P>, where δ(x) is the distance from x to δD.
    (3) Instead of the boundedness of K we prove the boundedness of an operator F from D to the outside of D with respect to suitable norms by using the maximal functions between D and the outside of D.
    We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer potentials.

  • クライン群の離散表現の集合の構造

    日本学術振興会  海外特別研究員

    Project Year :

    1997.10
    -
    1999.09
     

    松崎克彦

  • Deformation theory of Kleinian groups

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1997.04
    -
    1999.03
     

    WATANABE Hisako, MAEDA Michie, TANIZAKI Masahiko, MATSUZAKI Katsuhiko

     View Summary

    1. By geometric properties of corresponding a hyperbolic manifold we described the necessary conditions in order that the Hausdorff dimension of the limite of a n-dimensinal hyperbolic discrete group is less than n.
    2. We investigated the continuity of the Hausdorff dimensions as the Kleinian groups are deformed.
    3. We analized the structures of the set of discrete representations in the parameter space and proposed the method of analizing the structure of the space of quasi-Fuchsian groups by a holonomy map from the space of projections on a Riemann manifold.
    4. We obtained the results on the structural stability of Kleinian groups under small perturbations and on the equivalence between the algebraic topology and the Teichm_ller one in the space of quasiconformal deformations.
    5. We built the deformation theory of dynamical systems for entire functions on a ground of Teichm_ller space and found the fundamental properties of wan-dering domains and Baker domains, which rational functions don't have.
    6. We investigated a family of inlaid functions and showed the topological completeness of it. Further we found the Teichm_ller spaces of the Fatou components of those functions.

  • クライン群と複素力学系の研究

    お茶の水女子大学  科学研究費 奨励研究(A)

    Project Year :

    1996.04
    -
    1997.03
     

    松崎克彦

  • STUDY OF SELF-SIMILAR PROCESSES

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1996
    -
    1997
     

    KASAHARA Yuji, NARITA Kiyoko, MATSUZAKI Katsuhiko, YOSHIDA Hiroaki, KANEKO Akira, TAKEO Fukiko

     View Summary

    ・We studied the limiting processes for occupation times of fractional Brownian motion, which is a typical self-similar stochastic process. We see that the limiting process degenerates with the usual linear normalization but if consider the processes with the log-scale, we do have a meaningful process, which is the inverse of the so-called extremal process.
    ・We generalized the above result to a certain class of Gaussian processes, where the occupation time increases slowly.
    ・We studied the asymptotic behavior of the local times of fractional Brownian motion. As the index times the dimension approaches 1, we see that the one-dimensional marginal distributions of the local time at the origin converge to the exponentioal distribution, and furthermore, the processes, with a suitable time scale, converge to the inverse extremal process.
    ・The invariant set under a family of functions has self-similarity. We studied its topological aspects.

  • リーマン面のスペクトル理論

    日本学術振興会  特定国派遣研究員

    Project Year :

    1995.04
    -
    1995.09
     

    松崎克彦

  • 双曲的多様体の剛性と離散群のエルゴード性の研究

    東京工業大学  科学研究費 奨励研究(A)

    Project Year :

    1994.04
    -
    1995.03
     

    松崎克彦

  • 双曲的三次元多様体とクライン群

    東京工業大学  科学研究費 奨励研究(A)

    Project Year :

    1993.04
    -
    1994.03
     

    松崎克彦

  • 多様体上の複〓解析

    日本学術振興会  科学研究費助成事業

    Project Year :

    1992
     
     
     

    吹田 信之, 野田 洋二, 松崎 克彦, 志賀 啓成, 辻 元, 野口 潤次郎

     View Summary

    本研究において、代表者は、解析関数の特異性に関するRadsの定理を考察し、見通しのよい証明を得た。また解析関数のつくる関数空間について 志賀啓成は、軍位円枝で知られていた調和H_1空間とBMO空間の双対性関係を、Parabolie endを持つリーマン面に拡張した。さらに、有限な境界付リーマン面について、解析関数のH_1空間とBMO空間の双対性を考察し、さらに境界値関数が、リーマン面上のH_1関数の境界値となるための条件をみちびいた。松崎克彦は、双曲的離散群について、無限境球面上での作用が、エルゴード成分を持たないが、保存的にならない例を任意次元で構成した。また、クライン群について、幾何学的有限性、擬等角安定性、Bers写像の全射性の三つの性質の同値性を調べた。さらにフックス群が発散的であるための条件を、その正規部分群の性質で与え、またフックス群の作用のエルゴード成分が擬等角変形で保たれない例を構成した。この他タイヒミュラー空間の境界に関連する興味深い群を構成した。野田洋二は射影空間内の正則曲線の剛性定理を研究し、曲線が代数的に退化する様子を、除外集合を利用して考察し、いくつかの有用な結果を得、さらにそのような正則曲線の形を決定した。野口潤次郎は、双曲的多様体に関する2つのラング予想を解決し、これにより双曲的多様体への正則写像のモジュライ空間の構造を明らかにし、また同じ多様体に関する関数体上のMordell予想の解決を得た。さらに、双曲的幾何学と値分布理論に関する新しい関数の提起を行った。
    今後の展開としては、Rado型定理に現れる除外集合の必要性、ハーディ空間の数値け量の研究、解析写像の剛性、有限性の研究などが考えられる。

  • 複素構造の解析学的・幾何学的研究

    科学研究費助成事業(東京工業大学)  科学研究費助成事業(一般研究(B))

  • 多様体上の複素解析

    科学研究費助成事業(東京工業大学)  科学研究費助成事業(一般研究(C))

  • 複素多様体からの群の表現の解析

    科学研究費助成事業(東京工業大学) 

  • Study of various problems in mathematical physics

  • 幾何学的複素解析とポテンシャル論

    科学研究費助成事業(東京工業大学)  科学研究費助成事業(総合研究(A))

▼display all

Misc

  • 書評 L.V.アールフォルス(谷口雅彦訳) : 擬等角写像講義,数学クラシックス,29,丸善出版,2015年,168ページ

    松崎 克彦

    数学   72 ( 1 ) 94 - 98  2020  [Refereed]  [Invited]

    Book review, literature introduction, etc.  

    CiNii

  • 無限次元タイヒミュラー空間の問題 (複素幾何学の諸問題)

    松崎 克彦

    数理解析研究所講究録   1731   28 - 39  2011.03

    CiNii

  • Appendix G. The Denjoy-Wolff theorem

    Katsuhiko Matsuzaki

    Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906-1942, American Mthematical Society     303 - 305  2011  [Invited]

    Article, review, commentary, editorial, etc. (other)  

  • Appendix D. Kleinian groups

    Katsuhiko Matsuzaki

    Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906-1942, American Mthematical Society     295 - 296  2011  [Invited]

    Article, review, commentary, editorial, etc. (other)  

  • Structure theorem for holomorphic self-covers of Riemann surfaces and its applications (Infinite dimensional Teichmuller spaces and moduli spaces)

    FUJIKAWA Ege, MATSUZAKI Katsuhiko, TANIGUCHI Masahiko

    RIMS Kokyuroku Bessatsu   17   21 - 36  2010.06  [Refereed]

    CiNii

  • An averaging operator and non-separability of certain Banach spaces of holomorphic automorphic forms (Infinite dimensional Teichmuller spaces and moduli spaces)

    MATSUZAKI Katsuhiko

    RIMS Kokyuroku Bessatsu   17   65 - 72  2010.06  [Refereed]

    CiNii

  • Properties of asymptotically elliptic modular transformations of Teichmuller spaces (Infinite dimensional Teichmuller spaces and moduli spaces)

    MATSUZAKI Katsuhiko

    RIMS Kokyuroku Bessatsu   17   73 - 84  2010.06  [Refereed]

    CiNii

  • The projection of limit sets of modular groups on asymptotic Teichmüller spaces

    E. Fujikawa, K. Matsuzaki

    Proceedings of the 16th ICFIDCAA, Dongguk Univ., Daeyang Printing     86 - 92  2009

    Article, review, commentary, editorial, etc. (international conference proceedings)  

  • An example of self-covering of Riemann surface

    K. Matsuzaki, Y. Yabuki

    Proceedings of the 16th ICFIDCAA, Dongguk Univ., Daeyang Printing     86 - 92  2009

    Article, review, commentary, editorial, etc. (international conference proceedings)  

  • ポアンカレと非ユークリッド幾何学 (特集 ポアンカレ--知の巨人が放った創造性の至宝)

    松崎 克彦

    数理科学   46 ( 10 ) 18 - 24  2008.10

    CiNii

  • A remark on quadratic differentials vanishing at infinity(Complex Analysis and Geometry of Hyperbolic Spaces)

    MATSUZAKI KATSUHIKO

    RIMS Kokyuroku   1518   144 - 145  2006.10

    CiNii

  • The dynamics on Teichmuller spaces induced by holomorphic self-coverings(Complex Dynamics and its Related Fields)

    Fujikawa Ege, Matsuzaki Katsuhiko, Taniguchi Masahiko

    RIMS Kokyuroku   1494   44 - 48  2006.05

    CiNii

  • 微小時間の無限大/レビュー『博士の愛した数式』 (特集 美しき数式の世界)

    松崎 克彦

    数学セミナー   45 ( 2 ) 42 - 45  2006.02

    CiNii

  • 函数論的クライン群論が残したもの (特集 サーストン・プログラムと双曲幾何)

    松崎 克彦

    数学セミナー   44 ( 3 ) 22 - 25  2005.03

    CiNii

  • 擬対称写像とタイヒミュラーモジュラー群

    松崎 克彦

    総合講演・企画特別講演アブストラクト   2005 ( 0 ) 41 - 50  2005  [Invited]

    DOI CiNii

  • Dynamics of Teichmuller modular groups and general topology of moduli spaces : Announcement (Perspectives of Hyperbolic Spaces II)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1387   81 - 94  2004.07

    CiNii

  • An extension of the collar lemma (Perspectives of Hyperbolic Spaces)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1329   58 - 61  2003.06

    CiNii

  • The action of isotropy subgroups of the modular groups on infinite dimensional Teichmuller spaces (Hyperbolic Spaces and Discrete Groups II)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1270   84 - 87  2002.06

    CiNii

  • Locally connected tree-like invariant continua under Kleinian groups (Hyperbolic Spaces and Discrete Groups)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1223   31 - 32  2001.07

    CiNii

  • LOCAL GEOMETRIC FINITENESS OF KLEINIAN GROUPS (Hyperbolic Spaces and Related Topics II)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1163   42 - 45  2000.07

    CiNii

  • クライン群の力学系 : 極限集合のハウスドルフ次元

    松崎 克彦

    数学   51 ( 2 ) 142 - 160  1999.04  [Refereed]  [Invited]

    DOI CiNii

  • C. T. McMullen氏の業績

    松崎 克彦

    数学   51 ( 2 ) 186 - 188  1999.04  [Refereed]  [Invited]

    CiNii

  • A remark on the critical exponent of Kleinian groups (Analysis and Geometry of Hyperbolic Spaces)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1065   106 - 107  1998.10

    CiNii

  • クライン群の幾何学的収束と極限集合のハウスドルフ次元 (複素力学系の諸問題)

    松崎 克彦

    数理解析研究所講究録   1042   176 - 190  1998.04

    CiNii

  • Conditional stability of Kleinian groups

    松崎 克彦

    Science bulletin of Josai University, Special Issue   4   25 - 28  1998

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    NLA97 : Complex Dynamical Systems : The Second Symposium on Non-Linear Analysis and its Applications. / Edited by KIYOKO NISHIZAWA. 29-31 May 1997. The Conference Hall, MIZUTA Memorial Library Josai University.

    DOI CiNii

  • Stability of Kleinian groups(Analysis of Discrete Groups II)

    Matsuzaki Katsuhiko

    RIMS Kokyuroku   1022   87 - 92  1997.12

    CiNii

  • ユナボマ-の数学

    松崎 克彦

    数学セミナ-   36 ( 4 ) 44 - 46  1997.04

    CiNii

  • THE RATIO OF TWO NORMS OF QUADRATIC DIFFERENTIALS(Analysis of Discrete Groups)

    MATSUZAKI KATSUHIKO

    RIMS Kokyuroku   967   117 - 120  1996.10

    CiNii

  • Circle packing のリーマン写像への収束(Circle Packingの幾何学)

    松崎 克彦

    数理解析研究所講究録   893   24 - 35  1995.01

    CiNii

  • Circle Packing の変形空間のパラメーター(Circle Packingの幾何学)

    松崎 克彦

    数理解析研究所講究録   893   70 - 79  1995.01

    CiNii

  • SEVERAL CHARACTERIZATIONS OF FUCHSIAN GROUPS OF DIVERGENCE TYPE(Complex Analysis on Hyperbolic 3-Manifolds)

    MATSUZAKI KATSUHIKO

    数理解析研究所講究録   882   51 - 56  1994.08

    CiNii

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Syllabus

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Sub-affiliation

  • Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

Internal Special Research Projects

  • 離散群の指数増大度に関する不等式と剛性の研究

    2020   ヨハネス イェーリッシュ

     View Summary

     本研究では,双曲性をもつ離散群の指数増大度を,擬等角不変測度の Patterson-Sullivan 理論とマルコフ連鎖の群拡張の熱力学形式の理論から解明し,クライン群などに現れる増大度剛性と余増大度剛性の双対性に理論的背景を与えることを目的としていた.とくに増大度と余増大度の間のある不等式の証明をめざしたが,今年度の研究ではその予備的な考察までしか進まなかった. 並行しておこなったより古典的な双曲離散群に関連する研究として,クライン群の Myrberg 極限集合のハウスドルフ次元と無限生成ショットキー群で一意化されるリーマン面については具体的な結果が得られた.

  • タイヒミュラー空間上の不変計量の構成と応用

    2019   Huaying WEI

     View Summary

    (1)実軸上のVMOタイヒミュラー空間を構成する強対称写像について,それ自身および逆写像の一様連続性を仮定すればその全体は群構造をもち,また退化Carleson 測度を誘導するような上半平面上の擬等角写像に拡張することが証明された.(2)実軸上の漸近的等角写像のタイヒミュラー空間の概念を一般化し,区分的な対称写像による空間を普遍タイヒミュラー空間の閉部分空間として定式化した.これらの空間の増大列による普遍タイヒミュラー空間を補間する結果および商空間の構成を得た.計量については,商空間の複素構造を定義し,商フィンスラー計量を与えた.また,小林計量とタイヒミュラー計量の比較について,先行研究の方法では解決しない問題点を提示した. 

  • ケーリーグラフの等長変換群の収束指数と重みが変動する離散ラプラシアンのスペクトル

    2017   ヨハネス イェーリッシュ

     View Summary

    自由群のケーリーグラフへの部分群の等長的作用に関する収束指数と,商グラフ上の離散ラプラシアンのスペクトルの底との間には Grigorchuk の余増大公式という関係がある.同様の結果は,双曲空間に作用するクライン群の収束指数と双曲多様体上のラプラシアンに対しても Sullivan らにより証明されたが,共通する点は,ココンパクトな群の収束指数の1/2で相転移が起こることである.本研究では,自由群のケーリーグラフの辺の長さを変動させた場合にも,部分群の収束指数に依存して定まる重み付きの離散ラプラシアンに対して,そのスペクトルの底との間に余増大公式の一般化が証明され,収束指数の1/2での相転移も確かめられた.

  • 無限次元タイヒミュラー空間上の計量と等長変換群の研究

    2014  

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    (1) ヘルダー連続微分をもつ円周の微分同相写像のタイヒミュラー空間を定義し,ベルトラミ微分のノルムから誘導される位相と微分同相写像のノルムから誘導される位相が同値であることを示した.(2) 単位円板上の p 乗可積分タイヒミュラー空間にフィンスラー計量を定義し,完備性およびタイヒミュラー計量との関係を考察した.(3) それそれのタイヒミュラー空間の複素構造に関する双正則自己同相写像,計量に関する等長写像,および標準的な基点変換写像の間の関係についての問題を定式化した.

  • 離散群上の有界関数空間における幾何学的群論の新展開

    2011  

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     群論における Hopf の問題は,群の自己全射準同型が単射となる条件を問い,co-Hopf 問題は自己単射準同型が全射となる条件と問うている.本研究課題では自己共役に関する co-Hopf 問題について考察した.双曲空間に作用する等長変換群の離散部分群(クライン群)に関して既に得られていた結果を,より一般にグロモフ双曲空間の等長変換からなる離散群に対して拡張した.証明には双曲空間の無限遠境界の極限集合上の群作用で不変な擬等角測度を用いた.このような擬等角測度は Coornaert により導入されたもので,クライン群に関する Patterson-Sullivan 測度の一般化と考えられる.研究ではまず,Patterson-Sullivan 測度について成立していた結果を擬等角不変測度についても拡張することからはじめた.とくに群作用のエルゴード性と擬等角不変測度の一意性についての結果を整理した.さらに群に対して定義されるポアンカレ級数が収束指数次元において発散する場合(発散型),このような擬等角不変測度の強い意味での一意性が成り立つことを示せたことが議論の大きな展開を可能にした.証明の方法は Tukia のクライン群に関する同様の結果の証明に習い,発散型であれば conical な極限集合上で擬等角不変測度が正の測度をもつことを示した. 応用として次の2点が挙げられる.双曲群はそのケーリーグラフがグロモフ双曲空間となる群であり,上記の議論を直接適用できる.したがって双曲群の自己共役に関する co-Hopf 問題について新たな知見を加えることができる.別の応用としては,上記の証明の過程でしめされた次の命題の意義を考えることがある.「グロモフ双曲空間の等長変換からなる離散群が発散型であるとき,それを正規部分群として含む離散群もまた発散型で収束指数も一致する.」この命題は,自由群をはじめとして双曲群一般に対する正規部分群の収束指数に関する研究に大きく寄与する可能性をもつ.

  • 無限次元タイヒミュラー空間の不変部分空間の研究

    2010  

     View Summary

    無限次元タイヒミュラー空間に作用する写像類群の部分群とその不変部分空間の研究により,円周の同相写像群がメビウス群と共役になるための条件を与える問題に対して一定の成果をみた.普遍タイヒミュラー空間の写像類群は,円周の擬対称写像群と同一視できる.この場合,漸近的タイヒミュラー空間上のファイバーを不変にする部分群が対称写像群である.対称写像群の作用の固定点(不動点)を求める立場から上記の共役問題を考察した.Markovic による基本結果により,写像類群の部分群がタイヒミュラー空間に固定点をもつための必要十分条件は,軌道が有界であることがわかっている.よって有界軌道をもつ部分群に制限し,それが不変にする部分空間内に固定点をもつための条件を定式化した.以前の自身の結果で,対称写像群の部分群一般に対しては固定点の存在は保証されないことはわかっていた.本研究では,対称写像を境界値としてもつ単位円板の擬等角写像の歪曲係数に可積分条件を与え,それをみたす部分群を考えれば,対応する不変部分空間(具体的には可積分な正則2次微分の空間)に固定点をみつけられることに注目した.擬等角写像の歪曲係数の可積分条件は,対称写像自身の滑らかさの条件への対応をもつことが知られている.この関係を精査することにより,たとえば 1+1/2 階より大きい連続微分をもつ円周の微分同相写像群に対して,それがメビウス群と共役となるための条件を記述することが可能になった.この方法をさらに進めることにより,1階より大きな連続微分をもつ微分同相写像群の共役問題に関する予想の解決に向けて,前進が期待できる.今後の課題として継続して研究する予定である.

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