scholarly journals A variational principle for topological pressure for certain non-compact sets

2009 ◽  
Vol 80 (3) ◽  
pp. 585-602 ◽  
Author(s):  
Daniel Thompson
Keyword(s):  
Variational Principle ◽  
Topological Pressure ◽  
Compact Sets

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.

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.

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.

Tail pressure and the tail entropy function

2011 ◽  
Vol 32 (4) ◽  
pp. 1400-1417 ◽  
Author(s):  
YUAN LI ◽  
ERCAI CHEN ◽  
WEN-CHIAO CHENG
Keyword(s):  
Variational Principle ◽  
Invariant Measures ◽  
Direct Proof ◽  
Equilibrium States ◽  
Entropy Function ◽  
Topological Pressure

AbstractBurguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.

scholarly journals Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 309
Author(s):  
Xianfeng Ma ◽  
Zhongyue Wang ◽  
Hailin Tan
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Image Measure ◽  
Measure Preserving Transformations

A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.

The local variational principle of topological pressure for sub-additive potentials

2010 ◽  
Vol 73 (11) ◽  
pp. 3525-3536 ◽  
Author(s):  
Beimei Chen ◽  
Bangfeng Ding ◽  
Yongluo Cao
Keyword(s):  
Variational Principle ◽  
Topological Pressure

COSET PRESSURE WITH SUB-ADDITIVE POTENTIALS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350012 ◽  
Author(s):  
YUN ZHAO ◽  
WEN-CHIAO CHENG
Keyword(s):  
Variational Principle ◽  
Ergodic Theory ◽  
Topological Dynamics ◽  
Topological Pressure ◽  
Basic Properties ◽  
Separated Sets ◽  
Metric Group

The goal of this paper is to define the coset topological pressure for sub-additive potentials via separated sets on a compact metric group. Analogues of basic properties for topological pressure hold. This study also reveals a variational principle for the coset topological pressure. The process of the proof is quite similar to that of Cao, Feng and Huang's approximations, but the analysis needs more techniques of ergodic theory and topological dynamics.

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.

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