Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps

2008 ◽  
Vol 28 (2) ◽  
pp. 501-533 ◽  
Author(s):  
KRERLEY OLIVEIRA ◽  
MARCELO VIANA
Keyword(s):  
Equilibrium State ◽  
Large Class ◽  
Ergodic Theory ◽  
Gibbs Measure ◽  
Transfer Operator ◽  
Operator Approach ◽  
Hölder Continuous ◽  
Thermodynamical Formalism ◽  
Classical Notion ◽  
Positive Eigenfunction

AbstractWe develop a Ruelle–Perron–Fröbenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Hölder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, which is piecewise Hölder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here.

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.

Partial Hölder continuity of minimizers of functionals satisfying a VMO condition

2017 ◽  
Vol 10 (1) ◽  
pp. 83-110 ◽  
Author(s):  
Christopher S. Goodrich
Keyword(s):  
Large Class ◽  
Partial Regularity ◽  
Hölder Continuity ◽  
Full Measure ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Open Set ◽  
Principal Assumption ◽  
Class Of Functions

AbstractFor a bounded, open set${\Omega\hskip-0.569055pt\subseteq\hskip-0.569055pt\mathbb{R}^{n}}$we consider the partial regularity of vectorial minimizers${u\hskip-0.853583pt:\hskip-0.853583pt\Omega\hskip-0.853583pt\rightarrow\hskip-% 0.853583pt\mathbb{R}^{N}}$of the functional$u\mapsto\int_{\Omega}f(x,u,Du)\,dx,$where${f:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times n}\rightarrow\mathbb{R}}$. The principal assumption we make is thatfis asymptotically related to a function of the form${(x,u,\xi)\mapsto a(x,u)F(\xi)}$, whereFpossessesp-Uhlenbeck structure and the partial maps${x\mapsto a(x,\cdot\,)}$and${u\mapsto a(\,\cdot\,,u)}$are, respectively, of class VMO and${\mathcal{C}^{0}}$. We demonstrate that any minimizer${u\in W^{1,p}(\Omega)}$of this functional is Hölder continuous on an open set${\Omega_{0}}$of full measure. Finally, we show by means of an example that our asymptotic relatedness condition is very general and permits a large class of functions.

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 Conformal measure and decay of correlation for covering weighted systems

1998 ◽  
Vol 18 (6) ◽  
pp. 1399-1420 ◽  
Author(s):  
CARLANGELO LIVERANI ◽  
BENOIT SAUSSOL ◽  
SANDRO VAIENTI
Keyword(s):  
Invariant Measure ◽  
Equilibrium State ◽  
Large Class ◽  
Compact Set ◽  
Mixing Rate ◽  
Hilbert Metric ◽  
Frobenius Operator ◽  
Conformal Measure ◽  
Conformal Measures ◽  
Compactness Arguments

We show that for a large class of piecewise monotonic transformations on a totally ordered, compact set one can construct conformal measures and obtain the exponential mixing rate for the associated equilibrium state. The method is based on the study of the Perron–Frobenius operator. The conformal measure, the density of the invariant measure and the rate of mixing are deduced by using an appropriate Hilbert metric, without any compactness arguments, even in the case of a countable to one transformation.

scholarly journals Rates in almost sure invariance principle for quickly mixing dynamical systems

2019 ◽  
Vol 20 (01) ◽  
pp. 2050002
Author(s):  
C. Cuny ◽  
J. Dedecker ◽  
A. Korepanov ◽  
F. Merlevède
Keyword(s):  
Brownian Motion ◽  
Dynamical Systems ◽  
Large Class ◽  
Exponential Decay ◽  
Invariance Principle ◽  
Decay Of Correlations ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Exponential Decay Of Correlations ◽  
Stretched Exponential Decay

For a large class of quickly mixing dynamical systems, we prove that the error in the almost sure approximation with a Brownian motion is of order [Formula: see text] with [Formula: see text]. Specifically, we consider nonuniformly expanding maps with exponential and stretched exponential decay of correlations, with one-dimensional Hölder continuous observables.

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