scholarly journals Equilibrium measures for certain isometric extensions of Anosov systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1154-1167 ◽  
Author(s):  
RALF SPATZIER ◽  
DANIEL VISSCHER
Keyword(s):  
Equilibrium Measure ◽  
Closed Manifold ◽  
Unique Equilibrium ◽  
Hölder Continuous ◽  
Equilibrium Measures ◽  
Negatively Curved ◽  
Continuous Potential ◽  
Anosov Systems ◽  
Frame Flow

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. Brin and Gromov’s theorem on the ergodicity of frame flows follows as a corollary. Our methods also give a corresponding result for automorphisms of the Heisenberg manifold fibering over the torus.

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 Unique equilibrium states, large deviations and Lyapunov spectra for the Katok map

2020 ◽  
pp. 1-38
Author(s):  
TIANYU WANG
Keyword(s):  
Equilibrium State ◽  
Level Sets ◽  
Additional Condition ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Thermodynamic Formalism ◽  
Hölder Continuous ◽  
Lyapunov Spectra ◽  
Continuous Potential ◽  
Level 2

We study the thermodynamic formalism of a $C^{\infty }$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Hölder continuous potential with one additional condition, or geometric $t$ -potential $\unicode[STIX]{x1D711}_{t}$ with $t<1$ , the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$ . We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.

scholarly journals Smale endomorphisms over graph-directed Markov systems

2020 ◽  
pp. 1-34
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Equilibrium Measure ◽  
Parabolic Systems ◽  
Skew Product ◽  
Pointwise Dimension ◽  
Markov Systems ◽  
Large Classes ◽  
Equilibrium Measures ◽  
Skew Products ◽  
Natural Extensions ◽  
Induced Maps

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

scholarly journals Thermodynamics of the Katok map

2017 ◽  
Vol 39 (3) ◽  
pp. 764-794 ◽  
Author(s):  
Y. PESIN ◽  
S. SENTI ◽  
K. ZHANG
Keyword(s):  
Limit Theorem ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Existence Of Equilibrium ◽  
Equilibrium Measures ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Continuous Potential ◽  
Uniqueness Of Equilibrium ◽  
Unstable Direction

We effect the thermodynamical formalism for the non-uniformly hyperbolic $C^{\infty }$ map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2)110(3) 1979, 529–547]. It is a slow-down of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\unicode[STIX]{x1D711}_{t}=-t\log \mid df|_{E^{u}(x)}|$ for any $t\in (t_{0},\infty )$, $t\neq 1$, where $E^{u}(x)$ denotes the unstable direction. We show that $t_{0}$ tends to $-\infty$ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to $\unicode[STIX]{x1D711}_{t}$ for all values of $t\in (t_{0},1)$.

scholarly journals Large deviation principles for non-uniformly hyperbolic rational maps

2010 ◽  
Vol 31 (2) ◽  
pp. 321-349 ◽  
Author(s):  
HENRI COMMAN ◽  
JUAN RIVERA-LETELIER
Keyword(s):  
Large Deviation ◽  
Periodic Points ◽  
Rational Maps ◽  
Pressure Function ◽  
Hölder Continuous ◽  
Large Deviation Principles ◽  
Continuous Potential ◽  
Uniform Hyperbolicity ◽  
Level 2 ◽  
Unique Equilibrium State

AbstractWe show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called ‘Topological Collet–Eckmann’. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.

On the transfer operator for rational functions on the Riemann sphere

1996 ◽  
Vol 16 (2) ◽  
pp. 255-266 ◽  
Author(s):  
Manfred Denker ◽  
Feliks Przytycki ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Transfer Operator ◽  
Rational Functions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Correlation Integral ◽  
Hölder Continuous Functions ◽  
Conformal Measure

AbstractLet T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

scholarly journals Invariance of green equilibrium measure on the domain

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 593-600 ◽  
Author(s):  
Stamatis Pouliasis
Keyword(s):  
Level Set ◽  
Wide Class ◽  
Equilibrium Measure ◽  
The Other ◽  
Equilibrium Potential ◽  
Compact Set ◽  
Compact Sets ◽  
Equilibrium Measures ◽  
The One

We prove that the Green equilibrium measure and the Green equilibrium energy of a compact set K relative to the domains D and ? are the same if and only if D is nearly equal to ?, for a wide class of compact sets K. Also, we prove that equality of Green equilibrium measures arises if and only if the one domain is related with a level set of the Green equilibrium potential of K relative to the other domain.

scholarly journals Equilibrium measures on trees

2021 ◽  
Author(s):  
Nicola Arcozzi ◽  
Matteo Levi
Keyword(s):  
Equilibrium Measure ◽  
Equilibrium Measures ◽  
Local Degree ◽  
Infinite Tree ◽  
Special Case

AbstractWe give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For $$p=2$$ p = 2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph’s boundary in terms of square tilings of cylinders.

Thermodynamic formalism for countable Markov shifts

1999 ◽  
Vol 19 (6) ◽  
pp. 1565-1593 ◽  
Author(s):  
OMRI M. SARIG
Keyword(s):  
Equilibrium Measure ◽  
Thermodynamic Formalism ◽  
Finite Measure ◽  
Complete Characterization ◽  
Topological Pressure ◽  
Recurrent Functions ◽  
Equilibrium Measures ◽  
Positive Recurrence ◽  
Markov Shifts ◽  
Definition Of

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

scholarly journals Escape of mass and entropy for geodesic flows

2017 ◽  
Vol 39 (2) ◽  
pp. 446-473
Author(s):  
FELIPE RIQUELME ◽  
ANIBAL VELOZO
Keyword(s):  
Phase Transition ◽  
Geodesic Flow ◽  
Upper Semicontinuity ◽  
Geodesic Flows ◽  
Parabolic Subgroups ◽  
Hölder Continuous ◽  
Upper Semicontinuous ◽  
Negatively Curved ◽  
Geometrically Finite ◽  
Loss Of Mass

In this paper, we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure-theoretical entropy is upper semicontinuous when there is no loss of mass. In the case where mass is lost, the critical exponents of parabolic subgroups of the fundamental group have a significant meaning. More precisely, the failure of upper-semicontinuity of the entropy is determined by the maximal parabolic critical exponent. We also study the pressure of positive Hölder-continuous potentials going to zero through the cusps. We prove that the pressure map $t\mapsto P(tF)$ is differentiable until it undergoes a phase transition, after which it becomes constant. This description allows us, in particular, to compute the entropy of the geodesic flow at infinity.

Sign in / Sign up

Export Citation Format

Share Document