scholarly journals The K-property for some unique equilibrium states in flows and homeomorphisms

2021 ◽  
pp. 1-24
Author(s):  
BENJAMIN CALL
Keyword(s):  
Discrete Time ◽  
Equilibrium States ◽  
Unique Equilibrium ◽  
Decomposition Theory ◽  
Time Result ◽  
Pressure Gap ◽  
The Family ◽  
Flow Case

Abstract We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.

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 Discrete-time Cohen-Grossberg neural networks with transmission delays and impulses

2009 ◽  
Vol 43 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Sannay Mohamad ◽  
Haydar Akça ◽  
Valéry Covachev
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Exponential Stability ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Continuous Time ◽  
Global Exponential Stability ◽  
Exponential Convergence ◽  
Unique Equilibrium ◽  
Transmission Delays

Abstract A discrete-time analogue is formulated for an impulsive Cohen- -Grossberg neural network with transmission delay in a manner in which the global exponential stability characterisitics of a unique equilibrium point of the network are preserved. The formulation is based on extending the existing semidiscretization method that has been implemented for computer simulations of neural networks with linear stabilizing feedback terms. The exponential convergence in the p-norm of the analogue towards the unique equilibrium point is analysed by exploiting an appropriate Lyapunov sequence and properties of an M-matrix. The main result yields a Lyapunov exponent that involves the magnitude and frequency of the impulses. One can use the result for deriving the exponential stability of non-impulsive discrete-time neural networks, and also for simulating the exponential stability of impulsive and non-impulsive continuous-time networks.

Flows with Unique Equilibrium States

1977 ◽  
Vol 99 (3) ◽  
pp. 486 ◽  
Author(s):  
Ernesto Franco
Keyword(s):  
Equilibrium States ◽  
Unique Equilibrium

A NOVEL GLOBAL EXPONENTIAL STABILITY RESULT FOR DISCRETE-TIME CELLULAR NEURAL NETWORKS WITH VARIABLE DELAYS

2006 ◽  
Vol 16 (06) ◽  
pp. 467-472 ◽  
Author(s):  
QIANG ZHANG ◽  
XIAOPENG WEI ◽  
JIN XU
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Stability Condition ◽  
Global Exponential Stability ◽  
Stability Result ◽  
Cellular Neural Networks ◽  
Unique Equilibrium ◽  
Variable Delays

Global exponential stability is considered for a class of discrete-time cellular neural networks with variable delays. By employing a discrete Halanay inequality, a new result is presented ensuring global exponential stability of the unique equilibrium point of the networks. The result extends and improves the earlier publications due to the fact that it removes some restrictions on the delay. An example is given to illustrate the effectiveness of the global exponential stability condition provided here.

Geometric barycentres of invariant measures for circle maps

2001 ◽  
Vol 21 (2) ◽  
pp. 511-532 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Invariant Measures ◽  
Compact Convex ◽  
Invariant Probability Measure ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Expanding Map ◽  
The Family ◽  
Real Analytic ◽  
Two Parameter ◽  
Measure Of Maximal Entropy

For a continuous circle map T, define the barycentre of any T-invariant probability measure \mu to be b(\mu)=\int_{S^1} z\, d\mu(z). The set \Omega of all such barycentres is a compact convex subset of \mathbb{C}. If T is conjugate to a rational rotation via a Möbius map, we prove \Omega is a disc. For every piecewise-onto expanding map we prove that the barycentre set has non-empty interior. In this case, each interior point is the barycentre of many invariant measures, but we prove that amongst these there is a unique one which maximizes entropy, and that this measure belongs to a distinguished two-parameter family of equilibrium states. This family induces a real-analytic radial foliation of int(\Omega), centred around the barycentre of the global measure of maximal entropy, where each ray is the barycentre locus of some one-parameter section of the family. We explicitly compute these rays for two examples. While developing this framework we also answer a conjecture of Z. Coelho [6] regarding limits of sequences of equilibrium states.

Topological Wiener–Wintner ergodic theorems and a randomL2ergodic theorem

1996 ◽  
Vol 16 (1) ◽  
pp. 179-206 ◽  
Author(s):  
Peter Walters
Keyword(s):  
Ergodic Theorem ◽  
Equilibrium States ◽  
Ergodic Theorems ◽  
Unique Equilibrium ◽  
Ergodic Transformations ◽  
Finite Set ◽  
Uniquely Ergodic ◽  
Measure Preserving Transformations

AbstractWe give some topological ergodic theorems inspired by theWiener-Wintnerergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique equilibrium states for shift maps. These latter results give randomL2ergodic theorems for a finite set of commuting measure-preserving transformations.

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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