11 Topological pressure

2021 ◽  
pp. 349-370
Keyword(s):  
Topological Pressure

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.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

scholarly journals Coding sequence density estimation via topological pressure

2014 ◽  
Vol 70 (1-2) ◽  
pp. 45-69 ◽  
Author(s):  
David Koslicki ◽  
Daniel J. Thompson
Keyword(s):  
Density Estimation ◽  
Topological Pressure ◽  
Coding Sequence

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 Topological pressure via saddle points

2008 ◽  
Vol 360 (01) ◽  
pp. 545-562 ◽  
Author(s):  
Katrin Gelfert ◽  
Christian Wolf
Keyword(s):  
Saddle Points ◽  
Topological Pressure

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.

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