Entropy and variational principles for holonomic probabilities of IFS

Abstract

In this work we introduce an idempotent pressure to level-2 functions and its associated density entropy. All this is related to idempotent pressure functions which is the natural concept that corresponds to the meaning of probability in the level-2 max-plus context. In this general framework the equilibrium states, maximizing the variational principle, are not unique. We investigate the connections with the general convex pressure introduced recently to level-1 functions by Biś, Carvalho, Mendes and Varandas. Our general setting contemplates the dynamical and not dynamical framework. We also study a characterization of the density entropy in order to get an idempotent pressure invariant by dynamical systems acting on probabilities; this is therefore a level-2 result. We are able to produce idempotent pressure functions at level-2 which are invariant by the dynamics of the pushforward map via a form of Ruelle operator.

Duality results for iterated function systems with a general family of branches

“…The number e P (c) is equal to the spectral radius of L c if c is Lipschitz (related results can be founded in [13], [11]). This way of define entropy and pressure is related with the Legendre's transform.…”

“…Indeed, when d α = kdα, where k is a positive constant, we have L c, α = L c+log(k),α . For a countable or finite set X, the usual measure α(x) represents the summation over the branches of the weighted IFS (see [9], [13], [17]). Writing α(x) = i δ x i we get the transference operator…”

Quantum Stochastic Processes, Quantum Iterated Function Systems and Entropy

“…We will present a notion of entropy for "invariant" (or "stationary") measures with support on density matrices. This definition is obtained by adapting the reasoning described in [5], [12] and [13] to the present situation. The main idea is to define this concept via the Ruelle operator and to avoid the use of partitions.…”

“…The present definition of entropy is obtained by adapting the reasoning described in [5], [12] and [13] to the setting we present in this work. The main idea is to define this concept via the Ruelle operator and to avoid the use of partitions.…”

“…Using this definition one can consider maximal pressure density states. This formulation can be seen as a mini-max problem (see [5] [12] [13]). In [1] it is described some applications of the pressure problem.…”