scholarly journals On the uniqueness of equilibrium states for piecewise monotone mappings

1990 ◽  
Vol 97 (1) ◽  
pp. 27-36 ◽  
Author(s):  
Manfred Denker ◽  
Gerhard Keller ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium States ◽  
Monotone Mappings ◽  
Piecewise Monotone ◽  
Uniqueness Of Equilibrium

scholarly journals Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

2021 ◽  
pp. 1-43
Author(s):  
DOMINIC VECONI
Keyword(s):  
Thermodynamic Formalism ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Srb Measure ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Surface Homeomorphisms ◽  
Young Towers ◽  
Uniqueness Of Equilibrium ◽  
Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

scholarly journals Equilibrium states for a class of skew products

2019 ◽  
Vol 40 (11) ◽  
pp. 3030-3050
Author(s):  
MARIA CARVALHO ◽  
SEBASTIÁN A. PÉREZ
Keyword(s):  
Existence And Uniqueness ◽  
Global Analysis ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Axiom A ◽  
Skew Products ◽  
Pure Mathematics ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We consider skew products on $M\times \mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

scholarly journals Extendability of Equilibria of Nematic Polymers

2008 ◽  
Vol 2008 ◽  
pp. 1-10
Author(s):  
Hongyun Wang ◽  
Hong Zhou
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Jacobian Matrix ◽  
Intersection Point ◽  
Equilibrium States ◽  
Intermolecular Potential ◽  
Small Perturbations ◽  
Nematic Polymers ◽  
Folding Point ◽  
Uniqueness Of Equilibrium

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

scholarly journals Equilibrium states for Mañé diffeomorphisms

2018 ◽  
Vol 39 (9) ◽  
pp. 2433-2455 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
TODD FISHER ◽  
DANIEL J. THOMPSON
Keyword(s):  
Large Deviations ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Potential Functions ◽  
Equilibrium States ◽  
Unique Equilibrium ◽  
The Family ◽  
Natural Classes ◽  
Uniqueness Of Equilibrium

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

scholarly journals Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

2020 ◽  
Vol 181 (6) ◽  
pp. 2131-2149
Author(s):  
J.-R. Chazottes ◽  
J. Moles ◽  
F. Redig ◽  
E. Ugalde
Keyword(s):  
Equilibrium States ◽  
Lattice Systems ◽  
Uniqueness Of Equilibrium

scholarly journals On equilibrium states for partially hyperbolic horseshoes

2016 ◽  
Vol 38 (1) ◽  
pp. 301-335 ◽  
Author(s):  
I. RIOS ◽  
J. SIQUEIRA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Three Dimensional ◽  
Small Variation ◽  
Equilibrium States ◽  
Dimensional System ◽  
Hölder Continuous ◽  
The Family ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

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