Variational Principle For Topological Pressure Research Articles

Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

Hidden Gibbs measures on shift spaces over countable alphabets

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions, which we call irreducible countable sofic shifts. We show the variational principle for topological pressure for some sub-additive sequences with tempered variation on irreducible countable sofic shifts. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. Applications are given to some dimension problems and study of factors of (generalized) Gibbs measures on certain subshifts over countable alphabets.

Topological pressure and the variational principle for actions of sofic groups

AbstractWe introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

A relative local variational principle for topological pressure

A relative local variational principle for topological pressure

A relative local variational principle for topological pressure

We define the relative local topological pressure for any given factor map and open cover, and prove the relative local variational principle of this pressure. More precisely, for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover U of X, a continuous, real-valued function f on X and an S-invariant measure ν on Y, we show that the corresponding relative local pressure P(T, f, U, y) satisfies $$ \mathop {\sup }\limits_{\mu \in \mathcal{M}(X,T)} \left\{ {h_\mu (T,\mathcal{U}|Y) + \int_X {f(x)d\mu (x):\pi \mu = \nu } } \right\} = \int_Y {P(T,f,\mathcal{U},y)d\nu (y)} , $$ where \( \mathcal{M} \)(X, T) denotes the family of all T-invariant measures on X. Moreover, the supremum can be attained by a T-invariant measure.

A variational principle for topological pressure for certain non-compact sets

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $\varphi: X \mapsto \IR$ be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form \[ \{x \in X : \lim_{n \ra \infty} \frac{1}{n} \sum_{i = 0}^{n-1} \varphi (f^i (x)) = \alpha \}. \] Analogous results were previously known for topological entropy. As an application, we prove multifractal analysis results for the entropy spectrum of a suspension flow over a continuous map with specification and the dimension spectrum of certain non-uniformly expanding interval maps.