12 The variational principle and equilibrium states

2021 ◽  
pp. 371-386
Keyword(s):  
Variational Principle ◽  
Equilibrium States

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.

Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus

2009 ◽  
Vol 30 (5) ◽  
pp. 1503-1528 ◽  
Author(s):  
ERIC OLIVIER
Keyword(s):  
Variational Principle ◽  
Potential Function ◽  
Invariant Measures ◽  
Equilibrium States ◽  
Invariant Sets ◽  
Affine Invariant ◽  
Full Dimension ◽  
Compact Subsets

AbstractWe consider the variational principle for dimension on compact subsets of the 2-torus which are invariant under a non-conformal expanding diagonal endomorphism. Condition (H) ensures that the invariant measures with full dimension are the equilibrium states of some potential function. This result applies to the problem of uniqueness of the measure with full dimension on the sofic affine-invariant sets.

Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-27
Author(s):  
HUYI HU ◽  
WEISHENG WU ◽  
YUJUN ZHU
Keyword(s):  
Variational Principle ◽  
Continuous Function ◽  
Invariant Measures ◽  
Equilibrium States ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Gateaux Differentiability ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism ◽  
Differentiability Properties

Abstract Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphism f. We define the unstable pressure $P^{u}(f, \varphi )$ of f at a continuous function $\varphi $ via the dynamics of f on local unstable leaves. A variational principle for unstable pressure $P^{u}(f, \varphi )$ , which states that $P^{u}(f, \varphi )$ is the supremum of the sum of the unstable entropy and the integral of $\varphi $ taken over all invariant measures, is obtained. U-equilibrium states at which the supremum in the variational principle attains and their relation to Gibbs u-states are studied. Differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Fréchet differentiability and their relations to u-equilibrium states, are also considered.

Preimage pressure for random transformations

2009 ◽  
Vol 29 (5) ◽  
pp. 1669-1687 ◽  
Author(s):  
YUJUN ZHU ◽  
ZHIMING LI ◽  
XIAOHONG LI
Keyword(s):  
Variational Principle ◽  
Invariant Measures ◽  
Equilibrium States ◽  
Image Entropy

AbstractIn this paper, preimage pressure, which is based on the preimage structure of the system, is defined and studied for random transformations. We obtain analogs of many known results of preimage entropy and preimage pressure for deterministic cases in Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113] and Zeng et al [Pre-image pressure and invariant measures. Ergod. Th. & Dynam. Sys.27 (2007), 1037–1052]. In particular, a variational principle is given and some applications of preimage pressure, such as the investigation of the invariant measures and the equilibrium states, are obtained.

scholarly journals Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems

2020 ◽  
pp. 2150021
Author(s):  
Xinsheng Wang ◽  
Weisheng Wu ◽  
Yujun Zhu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Topological Entropy ◽  
Equilibrium States ◽  
Metric Entropy ◽  
Unstable Foliation ◽  
Hyperbolic Dynamical Systems ◽  
Partially Hyperbolic

Let [Formula: see text] be a [Formula: see text] random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of [Formula: see text] on the unstable foliation are introduced and investigated. A version of Shannon–McMillan–Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs [Formula: see text]-states are investigated.

Macroeconomic Equilibrium Theory in the Context of Methodological Problems of Contemporary Economic Science

2008 ◽  
pp. 77-88
Author(s):  
M. Likhachev
Keyword(s):  
Empirical Analysis ◽  
Equilibrium States ◽  
Equilibrium Theory ◽  
Economic Systems ◽  
Economic Science ◽  
The Real ◽  
Theoretical Status ◽  
Methodological Problems ◽  
Real Stability

The article is devoted to the analysis of methodological problems in using the conception of macroeconomic equilibrium in contemporary economics. The author considers theoretical status and relevance of equilibrium conception and discusses different areas and limits of applicability of the equilibrium theory. Special attention is paid to different epistemological criteria for this theory taking into account both empirical analysis of the real stability of economic systems and the problem of unobservability of equilibrium states.

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