scholarly journals Conformal deformation of equilibrium measures in normal random ensembles

2011 ◽  
Vol 44 (7) ◽  
pp. 075202 ◽  
Author(s):  
A M Veneziani ◽  
T Pereira ◽  
D H U Marchetti
Keyword(s):  
Conformal Deformation ◽  
Equilibrium Measures

scholarly journals BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS

2010 ◽  
Vol 21 (01) ◽  
pp. 77-115 ◽  
Author(s):  
ROBERT J. BERMAN
Keyword(s):  
Toeplitz Operators ◽  
Equilibrium Measure ◽  
Markov Property ◽  
General Setting ◽  
Space Structure ◽  
Tensor Power ◽  
Bergman Kernels ◽  
Equilibrium Measures ◽  
Holomorphic Sections ◽  
Positive Line Bundle

Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

scholarly journals Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

2014 ◽  
Vol 36 (1) ◽  
pp. 215-255 ◽  
Author(s):  
SAMUEL SENTI ◽  
HIROKI TAKAHASI
Keyword(s):  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Statistical Properties ◽  
Limit Set ◽  
Bifurcation Parameter ◽  
Large Interval ◽  
Equilibrium Measures ◽  
Uniform Hyperbolicity ◽  
Unstable Direction

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential$-t\log J^{u}$, where$t\in \mathbb{R}$is in a certain large interval and$J^{u}$denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

scholarly journals Symbolic dynamics for non-uniformly hyperbolic systems

2020 ◽  
pp. 1-68
Author(s):  
YURI LIMA
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Markov Partition ◽  
Equilibrium Measures ◽  
Markov Partitions ◽  
Recent Advances ◽  
Closed Orbits ◽  
Symbolic Extension ◽  
Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

scholarly journals A Polyakov Formula for Sectors

2017 ◽  
Vol 28 (2) ◽  
pp. 1773-1839 ◽  
Author(s):  
Clara L. Aldana ◽  
Julie Rowlett
Keyword(s):  
Heat Kernel ◽  
Explicit Expression ◽  
Unit Area ◽  
Logarithmic Singularity ◽  
Conformal Factor ◽  
Opening Angle ◽  
Conformal Deformation ◽  
Finite Area ◽  
Different Parts

Abstract We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw–Sommerfeld’s heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

scholarly journals Ergodic properties of equilibrium measures for smooth three dimensional flows

2016 ◽  
Vol 91 (1) ◽  
pp. 65-106 ◽  
Author(s):  
François Ledrappier ◽  
Yuri Lima ◽  
Omri Sarig
Keyword(s):  
Three Dimensional ◽  
Equilibrium Measures ◽  
Ergodic Properties

scholarly journals Ricci curvature of conformal deformation on compact 2-manifolds

2020 ◽  
Vol 19 (6) ◽  
pp. 3223-3231
Author(s):  
Yoon-Tae Jung ◽  
◽  
Soo-Young Lee ◽  
Eun-Hee Choi
Keyword(s):  
Ricci Curvature ◽  
Conformal Deformation

scholarly journals Hyperbolization of Euclidean Ornaments

10.37236/78 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Martin von Gagern ◽  
Jürgen Richter-Gebert
Keyword(s):  
Differential Geometry ◽  
Pattern Recognition ◽  
Hyperbolic Plane ◽  
Discrete Differential Geometry ◽  
Fundamental Region ◽  
Symmetry Groups ◽  
Symmetry Detection ◽  
Conformal Deformation ◽  
Optimal Resolution ◽  
Wallpaper Groups

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.

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