STOCHASTIC POTENTIALS OF INTERMITTENT MAPS

2020 ◽  
pp. 1-9
Author(s):  
HUAIBIN LI
Keyword(s):  
Nonempty Interior ◽  
Topological Pressure ◽  
Hölder Continuous ◽  
Continuous Potential

Consider an intermittent map  $f_{\unicode[STIX]{x1D705}}:[0,1]\rightarrow [0,1]$ and a Hölder continuous potential $\unicode[STIX]{x1D711}:[0,1]\rightarrow \mathbb{R}$ . We show that $\unicode[STIX]{x1D719}$ is stochastic for $f_{\unicode[STIX]{x1D705}}$ if and only if the topological pressure $P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711})$ satisfies $P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711}(0)>0$ . As a consequence, for each $\unicode[STIX]{x1D6FD}>0$ sufficiently small, the set of Hölder continuous potentials of exponent $\unicode[STIX]{x1D6FD}$ that are not stochastic for $f_{\unicode[STIX]{x1D705}}$ has nonempty interior in the space of all such potentials.

scholarly journals Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

2015 ◽  
Vol 37 (1) ◽  
pp. 79-102 ◽  
Author(s):  
THIAGO BOMFIM ◽  
PAULO VARANDAS
Keyword(s):  
Large Deviations ◽  
Gibbs Measures ◽  
Rate Function ◽  
Topological Pressure ◽  
Hölder Continuous ◽  
Continuous Potential ◽  
Hyperbolic Flows ◽  
Time Averages ◽  
Set Of Points ◽  
Level 2

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if$f$has an expanding repeller and$\unicode[STIX]{x1D719}$is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval$I\subset \mathbb{R}$can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

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 Equilibrium stability for non-uniformly hyperbolic systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2619-2642 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VANESSA RAMOS ◽  
JAQUELINE SIQUEIRA
Keyword(s):  
Hyperbolic Systems ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Equilibrium Stability ◽  
Hölder Continuous ◽  
Skew Products ◽  
Uniformly Hyperbolic Systems ◽  
Wide Family

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

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.

scholarly journals Induced topological pressure for countable state Markov shifts

2014 ◽  
Vol 14 (02) ◽  
pp. 1350016 ◽  
Author(s):  
Johannes Jaerisch ◽  
Marc Kesseböhmer ◽  
Sanaz Lamei
Keyword(s):  
Large Class ◽  
Group Structure ◽  
Topological Pressure ◽  
Hölder Continuous ◽  
Dynamical Group ◽  
Height Functions ◽  
Markov Shifts ◽  
Countable State ◽  
Set Up ◽  
Definition Of

We generalise Savchenko's definition of topological entropy for special flows over countable Markov shifts by considering the corresponding notion of topological pressure. For a large class of Hölder continuous height functions not necessarily bounded away from zero, this pressure can be expressed by our new notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words, and we are able to set up a variational principle in this context. Investigating the dependence of induced pressure on the subset of words, we give interesting new results connecting the Gurevič and the classical pressure with exhaustion principles for a large class of Markov shifts. In this context we consider dynamical group extensions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.

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.

Continuity of sub-additive topological pressure with matrix cocycles *

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 567-588
Author(s):  
Rui Zou ◽  
Yongluo Cao ◽  
Yun Zhao
Keyword(s):  
Value Function ◽  
General Setting ◽  
Singular Value ◽  
Topological Pressure ◽  
Finite Collection ◽  
Pressure Function ◽  
Hölder Continuous ◽  
Affine Maps ◽  
Affine Set ◽  
Funct Anal

Abstract Let A = {A 1, A 2, …, A k } be a finite collection of contracting affine maps, the corresponding pressure function P(A, s) plays the fundamental role in the study of dimension of self-affine sets. The zero of the pressure function always give the upper bound of the dimension of a self-affine set, and is exactly the dimension of ‘typical’ self-affine sets. In this paper, we consider an expanding base dynamical system, and establish the continuity of the pressure with the singular value function of a Hölder continuous matrix cocycle. This extends Feng and Shmerkin’s result in (Feng and Shmerkin 2014 Geom. Funct. Anal. 24 1101–1128) to a general setting.

scholarly journals On involution kernels and large deviations principles on $ \beta $-shifts

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Victor Vargas
Keyword(s):  
Large Deviations ◽  
Gibbs State ◽  
Natural Extension ◽  
Rate Function ◽  
Hölder Continuous ◽  
Large Deviations Principle ◽  
Ruelle Operator ◽  
The Family ◽  
In Series ◽  
Continuous Potential

<p style='text-indent:20px;'>Consider <inline-formula><tex-math id="M2">\begin{document}$ \beta &gt; 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lfloor \beta \rfloor $\end{document}</tex-math></inline-formula> its integer part. It is widely known that any real number <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in \Bigl[0, \frac{\lfloor \beta \rfloor}{\beta - 1}\Bigr] $\end{document}</tex-math></inline-formula> can be represented in base <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> using a development in series of the form <inline-formula><tex-math id="M6">\begin{document}$ \alpha = \sum_{n = 1}^\infty x_n\beta^{-n} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ x = (x_n)_{n \geq 1} $\end{document}</tex-math></inline-formula> is a sequence taking values into the alphabet <inline-formula><tex-math id="M8">\begin{document}$ \{0,\; ...\; ,\; \lfloor \beta \rfloor\} $\end{document}</tex-math></inline-formula>. The so called <inline-formula><tex-math id="M9">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-shift, denoted by <inline-formula><tex-math id="M10">\begin{document}$ \Sigma_\beta $\end{document}</tex-math></inline-formula>, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy <inline-formula><tex-math id="M11">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-expansion of <inline-formula><tex-math id="M12">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>. Fixing a Hölder continuous potential <inline-formula><tex-math id="M13">\begin{document}$ A $\end{document}</tex-math></inline-formula>, we show an explicit expression for the main eigenfunction of the Ruelle operator <inline-formula><tex-math id="M14">\begin{document}$ \psi_A $\end{document}</tex-math></inline-formula>, in order to obtain a natural extension to the bilateral <inline-formula><tex-math id="M15">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-shift of its corresponding Gibbs state <inline-formula><tex-math id="M16">\begin{document}$ \mu_A $\end{document}</tex-math></inline-formula>. Our main goal here is to prove a first level large deviations principle for the family <inline-formula><tex-math id="M17">\begin{document}$ (\mu_{tA})_{t&gt;1} $\end{document}</tex-math></inline-formula> with a rate function <inline-formula><tex-math id="M18">\begin{document}$ I $\end{document}</tex-math></inline-formula> attaining its maximum value on the union of the supports of all the maximizing measures of <inline-formula><tex-math id="M19">\begin{document}$ A $\end{document}</tex-math></inline-formula>. The above is proved through a technique using the representation of <inline-formula><tex-math id="M20">\begin{document}$ \Sigma_\beta $\end{document}</tex-math></inline-formula> and its bilateral extension <inline-formula><tex-math id="M21">\begin{document}$ \widehat{\Sigma_\beta} $\end{document}</tex-math></inline-formula> in terms of the quasi-greedy <inline-formula><tex-math id="M22">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-expansion of <inline-formula><tex-math id="M23">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> and the so called involution kernel associated to the potential <inline-formula><tex-math id="M24">\begin{document}$ A $\end{document}</tex-math></inline-formula>.</p>

Thermodynamic formalism for a modified shift map

2014 ◽  
Vol 14 (02) ◽  
pp. 1350020 ◽  
Author(s):  
Stephen Muir ◽  
Mariusz Urbański
Keyword(s):  
A Priori ◽  
Transfer Operator ◽  
Thermodynamic Formalism ◽  
Lattice Gases ◽  
Hölder Continuous ◽  
Full Generality ◽  
Counting Measure ◽  
Shift Map ◽  
Continuous Potential ◽  
Shift Action

We introduce a transfer operator and use it to prove some theorems of a classical flavor from thermodynamic formalism (including existence and uniqueness of appropriately defined Gibbs states and equilibrium states for potential functions satisfying Dini's condition and stochastic laws for Hölder continuous potential and observable functions) in a novel setting: the "alphabet" E is a compact metric space equipped with an a priori probability measure ν and an endomorphism T. The "modified shift map" S is defined on the product space Eℕ by the rule (x1x2x3…) ↦ (T(x2)x3…). The greatest novelty is found in the variational principle, where a term must be added to the entropy to reflect the transformation of the first coordinate by T after shifting. Our motivation is that this system, in its full generality, cannot be treated by the existing methods of either rigorous statistical mechanics of lattice gases (where only the true shift action is used) or dynamical systems theory (where the a priori measure is always implicitly taken to be the counting measure).

Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms

2021 ◽  
Author(s):  
Svetlana Jitomirskaya ◽  
Saša Kocić
Keyword(s):  
Type Theorem ◽  
Schrödinger Operators ◽  
Dynamical Analysis ◽  
Schrodinger Operators ◽  
Hölder Continuous ◽  
Rotation Numbers ◽  
Continuous Potential ◽  
Circle Diffeomorphism ◽  
Almost All ◽  
Circle Diffeomorphisms

Abstract We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s ${{\mathcal{H}}}$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $C^{1+BV}$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $\mu $, and for $\mu $-almost all $x\in{{\mathbb{T}}}^1$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $V$.

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