scholarly journals Entropy for hyperbolic Riemann surface laminations I

2014 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Viet-Anh Nguyen ◽  
Nessim Sibony
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Careful Analysis ◽  
Probability Measures ◽  
Metric Entropy ◽  
Hölder Continuous ◽  
Poincaré Metric ◽  
Singular Foliations ◽  
Open Questions ◽  
Hyperbolic Riemann Surface

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.

scholarly journals Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

1974 ◽  
Vol 53 ◽  
pp. 141-155 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Banach Spaces ◽  
Riemann Surfaces ◽  
Space Structure ◽  
Bounded Solutions ◽  
Linear Isometry ◽  
Hölder Continuous ◽  
Hyperbolic Riemann Surfaces ◽  
Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

scholarly journals Entropy for hyperbolic Riemann surface laminations II

2014 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Viet-Anh Nguyen ◽  
Nessim Sibony
Keyword(s):  
Riemann Surface ◽  
Singular Point ◽  
Modulus Of Continuity ◽  
Riemann Surfaces ◽  
Complex Surface ◽  
Higher Dimension ◽  
Poincaré Metric ◽  
Local Setting ◽  
Compact Complex Surface ◽  
Hyperbolic Riemann Surface

This chapter studies Riemann surface foliations with tame singular points. It shows that the hyperbolic entropy of a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface is finite. The chapter then proves the finiteness of the entropy in the local setting near a singular point in any dimension, using a division of a neighborhood of a singular point into adapted cells. Next, the chapter estimates the modulus of continuity for the Poincaré metric along the leaves of the foliation, using notion of conformally (R,δ‎)-close maps. The estimate holds for foliations on manifolds of higher dimension.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

scholarly journals Limits of canonical forms on towers of Riemann surfaces

2020 ◽  
Vol 2020 (764) ◽  
pp. 287-304
Author(s):  
Hyungryul Baik ◽  
Farbod Shokrieh ◽  
Chenxi Wu
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Type Theorem ◽  
Universal Cover ◽  
Canonical Forms ◽  
Classical Version ◽  
Galois Cover ◽  
Hyperbolic Form ◽  
Galois Covers ◽  
Hyperbolic Riemann Surface

AbstractWe prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

scholarly journals The geometry of measured geodesic laminations and measured train tracks

1989 ◽  
Vol 9 (3) ◽  
pp. 587-604 ◽  
Author(s):  
Howard Weiss
Keyword(s):  
Riemann Surface ◽  
Linear Model ◽  
Tangent Space ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Hölder Continuous ◽  
Geodesic Laminations ◽  
Non Linear ◽  
Train Tracks ◽  
Hyperbolic Riemann Surface

AbstractThurston and Kerckhoff have shown that the space of measured geodesic laminations on a hyperbolic Riemann surface serves as a non-linear model of the tangent space to Teichmüller space at the surface. In this paper we show that the natural map between these manifolds has stronger than Hölder continuous regularity.

Dynamics of holomorphic correspondences on Riemann surfaces

2020 ◽  
Vol 31 (05) ◽  
pp. 2050036 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Lucas Kaufmann ◽  
Hao Wu
Keyword(s):  
Riemann Surface ◽  
Topological Degree ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Critical Values ◽  
Probability Measures ◽  
Global Dynamics ◽  
Forward Orbit ◽  
Holomorphic Correspondences

We study the global dynamics of holomorphic correspondences [Formula: see text] on a compact Riemann surface [Formula: see text] in the case, so far not well understood, where [Formula: see text] and [Formula: see text] have the same topological degree. In the absence of a mild and necessary obstruction that we call weak modularity, [Formula: see text] admits two canonical probability measures [Formula: see text] and [Formula: see text] which are invariant by [Formula: see text] and [Formula: see text] respectively. If the critical values of [Formula: see text] (respectively, [Formula: see text]) are not periodic, the backward (respectively, forward) orbit of any point [Formula: see text] equidistributes towards [Formula: see text] (respectively, [Formula: see text]), uniformly in [Formula: see text] and exponentially fast.

scholarly journals Canonical isomorphisms of energy finite solutions of Δu = Pu on open Riemann surfaces

1975 ◽  
Vol 56 ◽  
pp. 79-84 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Continuous Function ◽  
Linear Space ◽  
Riemann Surfaces ◽  
Second Order ◽  
Local Parameter ◽  
Hölder Continuous

We call a second order differential P(z)dxdy on a Riemann surface R a density if it is not identically zero and P(z) is a nonnegative Hölder continuous function of the local parameter z = x + iy in each parametric disk. To each density P on R we associate the linear space P(R) of C2 solutions of the equation Δu(z) = P(z)u(z) invariantly defined on R. We also consider subspaces PX(R) of P(R) consisting of solutions with certain boundedness properties X.

Fourier expansion along geodesics on Riemann surfaces

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Anton Deitmar
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Fourier Expansion ◽  
Summation Formula ◽  
Growth Estimate ◽  
Hyperbolic Riemann Surface

AbstractFor an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.

scholarly journals Analytic Functions on Some Riemann Surfaces, II

1963 ◽  
Vol 23 ◽  
pp. 153-164 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Harmonic Measure ◽  
Analytic Functions ◽  
Boundary Point ◽  
Riemann Surfaces ◽  
Positive Answer ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surface

In their paper [12], Toda and the author have concerned themselves in the followingTheorem of Kuramochi. Let R be a hyperbolic Riemann surface of the class OHB(OHD, resp.). Then, for any compact subset K of R such that R−K is connected, R−K as an open Riemann surface belongs to the class OAB(OAD, resp.) (Kuramochi [4]).They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.

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