Entropy, pressure, ground states and calibrated sub-actions for linear dynamics

Abstract

Denote by $X$ a Banach space and by $T:X\to X$ a bounded linear operator with non-trivial kernel satisfying suitable conditions. We consider the concepts of entropy - for $T$-invariant probability measures - and pressure for Hölder continuous potentials. We also prove the existence of ground states (the limit when temperature goes to zero) associated with such class of potentials when the Banach space $X$ is equipped with a Schauder basis. We produce an example concerning weighted shift operators defined on the Banach spaces $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$, $1\le p<+\infty$, where our results do apply. In addition, we prove the existence of calibrated sub-actions when the potential satisfies certain regularity conditions using properties of the so-called Mañé potential. We also exhibit examples of selection at zero temperature and explicit sub-actions in the class of Hölder continuous potentials.

References in corpus (18)

• On the zero-temperature limit of Gibbs states
• Ergodic optimization in dynamical systems
• Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature
• Expansivity and Shadowing in Linear Dynamics
• Chaotic temperature dependence at zero temperature
• On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection
• Ergodic optimization, zero temperature limits and the max-plus algebra
• Zero-temperature phase diagram for double-well type potentials in the summable variation class
• Linear Dynamical Systems
• Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts
• Negative Entropy, Zero temperature and stationary Markov Chains on the interval
• Thermodynamic Formalism for Topological Markov Chains on Borel Standard Spaces
• On the Lyapunov spectrum of relative transfer operators
• Shadowing and structural stability in linear dynamical systems
• Gibbs States and Gibbsian Specifications on the space ${\mathbb{R}}^{\mathbb{N}}$
• The Ruelle operator for symmetric $\beta$-shifts
• Peierls barrier for countable Markov shifts
• Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

Cited by in corpus (2)

• Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism
• Existence of the zero temperature limit of equilibrium states on topologically transitive countable Markov shifts