scholarly journals Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

2012 ◽  
Vol 33 (3) ◽  
pp. 831-850 ◽  
Author(s):  
YONGLUO CAO ◽  
HUYI HU ◽  
YUN ZHAO
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Topological Pressure ◽  
Compact Sets ◽  
Equivalent Definition ◽  
Ergodic Measures ◽  
Nonadditive Measure ◽  
Definition Of ◽  
Conformal Repeller

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

scholarly journals A thermodynamic definition of topological pressure for non-compact sets

2010 ◽  
Vol 31 (2) ◽  
pp. 527-547 ◽  
Author(s):  
DANIEL J. THOMPSON
Keyword(s):  
Lyapunov Exponent ◽  
Variational Principle ◽  
Equilibrium State ◽  
Metric Space ◽  
Level Sets ◽  
Metric Spaces ◽  
Topological Pressure ◽  
Compact Sets ◽  
Alternative Definition ◽  
Definition Of

AbstractWe give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

Entropy of a Flow on Non-compact Sets

2012 ◽  
Vol 19 (02) ◽  
pp. 1250015 ◽  
Author(s):  
Jinghua Shen ◽  
Yun Zhao
Keyword(s):  
Variational Principle ◽  
Topological Entropy ◽  
Compact Sets ◽  
Entropy Formula ◽  
The One ◽  
Definition Of

Based on the theory of Carathéodory structure, this paper introduces the topological entropy of a flow on non-compact sets. Moreover, we introduce the definition of measure-theoretic entropy of a flow. It is shown that this entropy is equivalent to the one defined by Sun in [10]. The variational principle between topological entropy and measure-theoretic entropy of a flow is established. We also get the Brin-Katok's entropy formula for a flow.

APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

2011 ◽  
Vol 11 (04) ◽  
pp. 643-679 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Sierpinski Carpet ◽  
3 Dimensional ◽  
Sofic Shift ◽  
Full Shift ◽  
Sierpiński Carpet ◽  
Invariant Ergodic Measure

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

On the joint continuity of topological pressures for sub-additive singular-valued potentials

2021 ◽  
pp. 2150042
Author(s):  
Congcong Qu ◽  
Lan Xu
Keyword(s):  
Variational Principle ◽  
Upper Bound ◽  
Topological Pressure ◽  
Joint Continuity ◽  
Conformal Repeller

Given a non-conformal repeller [Formula: see text] of a [Formula: see text] map [Formula: see text], we give a variational principle of a dimensional upper bound of the non-conformal repellers. If [Formula: see text] is [Formula: see text] then we prove the joint continuity of topological pressure for sub-additive singular-valued potentials on maps and parameters.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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.

VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

2009 ◽  
Vol 09 (02) ◽  
pp. 205-215 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

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