Ergodic Theory – Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps

2021 ◽  
Author(s):  
Mariusz Urbański ◽  
Mario Roy ◽  
Sara Munday
Keyword(s):  
Ergodic Theory ◽  
Symbolic Dynamics ◽  
Thermodynamic Formalism ◽  
Expanding Maps

scholarly journals Symbolic dynamics in mean dimension theory

2020 ◽  
pp. 1-19
Author(s):  
MAO SHINODA ◽  
MASAKI TSUKAMOTO
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Symbolic Dynamics ◽  
Rate Distortion ◽  
Dimension Theory ◽  
Minimal Sets ◽  
Mean Dimension ◽  
The Mean

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$ -subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$ -subshifts with respect to a subaction of $\mathbb{Z}$ . The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$ -subshifts in terms of Kolmogorov–Sinai entropy.

Zeta functions for certain non-hyperbolic systems and topological Markov approximations

1998 ◽  
Vol 18 (6) ◽  
pp. 1589-1612 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Gibbs Measure ◽  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Thermodynamic Formalism ◽  
Zeta Functions ◽  
Product Formula ◽  
Markov Approximation ◽  
Weighted Functions ◽  
Countable State

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.

Invariant measures of full dimension for some expanding maps

1997 ◽  
Vol 17 (1) ◽  
pp. 147-167 ◽  
Author(s):  
DIMITRIOS GATZOURAS ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Large Class ◽  
Open Problem ◽  
Invariant Measures ◽  
Thermodynamic Formalism ◽  
Invariant Set ◽  
Invariant Sets ◽  
Expanding Maps ◽  
Full Dimension

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

Singular perturbation of symbolic dynamics via thermodynamic formalism

2008 ◽  
Vol 28 (4) ◽  
pp. 1261-1289 ◽  
Author(s):  
TAKEHIKO MORITA ◽  
HARUYOSHI TANAKA
Keyword(s):  
Singular Perturbation ◽  
Finite Type ◽  
Symbolic Dynamics ◽  
Thermodynamic Formalism ◽  
Unperturbed System ◽  
Large Parameter ◽  
Subshift Of Finite Type ◽  
Perturbed System ◽  
Perturbed Systems

AbstractWe consider singular perturbation of a mixing subshift of finite type by means of thermodynamic formalism. In our formulation, the perturbed systems are described by a family of potentials {Φ(α,⋅)} with large parameter α on a fixed subshift of finite type, and the original (unperturbed) system is characterized as the system at infinity obtained by collapsing the perturbed system upon taking $\alpha \to \infty $. We apply our formulation to the collapse of cookie-cutter systems and dispersing open billiards.

A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems

1996 ◽  
Vol 16 (5) ◽  
pp. 871-927 ◽  
Author(s):  
Luis M. Barreira
Keyword(s):  
Symbolic Dynamics ◽  
Broad Class ◽  
Thermodynamic Formalism ◽  
Upper Bounds ◽  
Dimension Theory ◽  
Lower And Upper Bounds ◽  
Closed Sets ◽  
Hyperbolic Dynamical Systems ◽  
Hyperbolic Sets ◽  
Additive Version

AbstractA non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they can be coded by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

MULTIFRACTAL DESCRIPTION OF THE STATISTICAL EQUILIBRIUM OF CHAOTIC DYNAMICAL SYSTEMS

1989 ◽  
Vol 03 (06) ◽  
pp. 821-852 ◽  
Author(s):  
JOSEPH L. McCAULEY
Keyword(s):  
Information Flow ◽  
Symbolic Dynamics ◽  
Chaotic Maps ◽  
Thermodynamic Formalism ◽  
Deterministic Chaos ◽  
Statistical Equilibrium ◽  
Coarse Grained ◽  
Matrix Formulation ◽  
Liapunov Exponents ◽  
Time Average

In this review, we apply the method of backward iteration to chaotic maps of the interval to illustrate how both the f(α) spectrum and its underlying statistical mechanics follow directly from the dynamics in statistical equilibrium. The sizes of intervals in the coarse-grained phase space are expressed directly in terms of finite-time average Liapunov exponents, representing the reverse of the information flow that is the underlying cause of deterministic chaos. The transfer matrix formulation follows directly from the method of backward iteration when addresses generated from symbolic dynamics are assigned to tree-branches. The inverse temperature is interpreted in terms of classes of initial data of the dynamical system. Finally, the usefulness of the thermodynamic formalism is illustrated by showing how the pore distribution of sandstone can be modeled by a certain two-scale Cantor set on an octal tree.

Uniformity of Lyapunov exponents for non-invertible matrices

2019 ◽  
Vol 40 (9) ◽  
pp. 2399-2433
Author(s):  
DE-JUN FENG ◽  
CHIU-HONG LO ◽  
SHUANG SHEN
Keyword(s):  
Lyapunov Exponents ◽  
Symbolic Dynamics ◽  
Absolute Continuity ◽  
Thermodynamic Formalism ◽  
Invariant Sets ◽  
Affine Invariant ◽  
Matrix Products ◽  
Self Similar ◽  
The Absolute ◽  
Invertible Matrices

Let $\mathbf{M}=(M_{1},\ldots ,M_{k})$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether $\mathbf{M}$ possesses the following property: there exist two constants $\unicode[STIX]{x1D706}\in \mathbb{R}$ and $C>0$ such that for any $n\in \mathbb{N}$ and any $i_{1},\ldots ,i_{n}\in \{1,\ldots ,k\}$, either $M_{i_{1}}\cdots M_{i_{n}}=\mathbf{0}$ or $C^{-1}e^{\unicode[STIX]{x1D706}n}\leq \Vert M_{i_{1}}\cdots M_{i_{n}}\Vert \leq Ce^{\unicode[STIX]{x1D706}n}$, where $\Vert \cdot \Vert$ is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on $\mathbb{R}$, the absolute continuity of certain self-affine measures in $\mathbb{R}^{d}$ and the dimensional regularity of a class of sofic affine-invariant sets in the plane.

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