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.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Thermodynamic formalism for certain nonhyperbolic maps

1999 ◽  
Vol 19 (5) ◽  
pp. 1365-1378 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Number Theory ◽  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Periodic Points ◽  
Two Dimensional ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Conformal Measures ◽  
One Dimensional Maps

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

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 Ergodic optimization in dynamical systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2593-2618 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Model Problem ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Ergodic Optimization ◽  
Maximizing Measure ◽  
Time Average

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called$f$-maximizing if the time average of the real-valued function$f$along the orbit is larger than along all other orbits, and an invariant probability measure is called$f$-maximizing if it gives$f$a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

scholarly journals Equilibrium states for Mañé diffeomorphisms

2018 ◽  
Vol 39 (9) ◽  
pp. 2433-2455 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
TODD FISHER ◽  
DANIEL J. THOMPSON
Keyword(s):  
Large Deviations ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Potential Functions ◽  
Equilibrium States ◽  
Unique Equilibrium ◽  
The Family ◽  
Natural Classes ◽  
Uniqueness Of Equilibrium

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

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 Lyapunov spectrum for Hénon-like maps at the first bifurcation

2016 ◽  
Vol 38 (3) ◽  
pp. 1168-1200
Author(s):  
HIROKI TAKAHASI
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Lyapunov Spectrum ◽  
Probability Measures ◽  
Limit Set ◽  
Uniform Hyperbolicity ◽  
Partial Description ◽  
Set Of Points

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

scholarly journals The boundary of hyperbolicity for Hénon-like families

2008 ◽  
Vol 28 (4) ◽  
pp. 1049-1080 ◽  
Author(s):  
YONGLUO CAO ◽  
STEFANO LUZZATTO ◽  
ISABEL RIOS
Keyword(s):  
Bifurcation Parameter ◽  
Uniform Hyperbolicity ◽  
Parameter Values

AbstractWe consider C2 Hénon-like families of diffeomorphisms of $ \mathbb R^{2} $ and study the boundary of the region of parameter values for which the non-wandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to the bifurcation parameter and even, to some extent, at the bifurcation parameter.

NOTE ON ERGODIC CHAOS IN THE RSS MODEL

2012 ◽  
Vol 17 (7) ◽  
pp. 1519-1524 ◽  
Author(s):  
Paulo S. A. Sousa
Keyword(s):  
Probability Measure ◽  
Invariant Probability Measure ◽  
Statistical Properties ◽  
Typical Trajectory

We show the possibility of ergodic chaos in the RSS model, due to Robinson, Solow, and Srinivasan. Moreover, under a relevant parametric regime, we analytically characterize the unique invariant probability measure that describes the statistical properties of a typical trajectory of capital stocks.

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