scholarly journals Ergodic properties of equilibrium states

1981 ◽  
Vol 80 (4) ◽  
pp. 477-483 ◽  
Author(s):  
Joseph Slawny
Keyword(s):  
Equilibrium States ◽  
Ergodic Properties

scholarly journals Equilibrium states for piecewise monotonic transformations

1982 ◽  
Vol 2 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Franz Hofbauer ◽  
Gerhard Keller
Keyword(s):  
Periodic Orbits ◽  
Bounded Variation ◽  
Critical Points ◽  
Equilibrium States ◽  
Absolutely Continuous ◽  
Specification Property ◽  
Ergodic Properties ◽  
Piecewise Monotonic

AbstractWe show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:(i) sup φ — inf φ <htop(T) and φ is of bounded variation.(ii) φ satisfies a variation condition and T has a local specification property.(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

scholarly journals Equilibrium states for partially hyperbolic horseshoes

2010 ◽  
Vol 31 (1) ◽  
pp. 179-195 ◽  
Author(s):  
R. LEPLAIDEUR ◽  
K. OLIVEIRA ◽  
I. RIOS
Keyword(s):  
Invariant Measures ◽  
Ergodic Measure ◽  
Equilibrium States ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Ergodic Properties ◽  
Central Direction ◽  
Positive Exponent ◽  
Continuous Potential ◽  
Partially Hyperbolic

AbstractWe study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 433–474]. These maps have a one-dimensional center direction Ec, and are at the boundary of the (uniformly) hyperbolic diffeomorphisms (they are constructed bifurcating hyperbolic horseshoes via heterodimensional cycles). We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the exception of one ergodic measure with positive exponent. As a consequence, we obtain the existence of equilibrium states for any continuous potential. We also prove that there exists a phase transition for the smooth family of potentials given by ϕt=t log ∣DF∣Ec∣.

scholarly journals A necessary and sufficient condition for a matrix equilibrium state to be mixing

2017 ◽  
Vol 39 (8) ◽  
pp. 2223-2234 ◽  
Author(s):  
IAN D. MORRIS
Keyword(s):  
Equilibrium State ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hölder Continuous ◽  
Ergodic Properties ◽  
Bernoulli Measure ◽  
Necessary And Sufficient

Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to Hölder-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved by Feng and Käenmäki [Equilibrium states of the pressure function for products of matrices.Discrete Contin. Dyn. Syst.30(3) (2011), 699–708]. In a recent article [Ergodic properties of matrix equilibrium states.Ergod. Th. & Dynam. Sys.(2017), to appear] the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli measure with respect to the natural partition, in terms of the algebraic properties of the semigroup generated by the matrices. Necessary and sufficient conditions were successfully established for the latter two properties, but only a sufficient condition for mixing was given. The purpose of this note is to complete that investigation by giving a necessary and sufficient condition for a matrix equilibrium state to be mixing.

scholarly journals Ergodic properties of matrix equilibrium states

2017 ◽  
Vol 38 (6) ◽  
pp. 2295-2320 ◽  
Author(s):  
IAN D. MORRIS
Keyword(s):  
Equilibrium State ◽  
Lyapunov Exponents ◽  
Equilibrium States ◽  
Invariant Equilibrium ◽  
Sufficient Condition ◽  
Ergodic Properties ◽  
General Sufficient Condition ◽  
Algebraic Properties ◽  
Bernoulli Measure ◽  
Zero Entropy

Given a finite irreducible set of real$d\times d$matrices$A_{1},\ldots ,A_{M}$and a real parameter$s>0$, there exists a unique shift-invariant equilibrium state on$\{1,\ldots ,M\}^{\mathbb{N}}$associated to$(A_{1},\ldots ,A_{M},s)$. In this paper we characterize the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterize when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by$A_{1},\ldots ,A_{M}$, and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by Kusuoka are explored in an appendix.

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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