Eduardo Garibaldi

On the Aubry-Mather theory for symbolic dynamics

We propose a new model of ergodic optimization for expanding dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In other contexts (for twist maps, for instance), this property appears in a crucial way. A version of the Aubry-Mather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the two-sided shift. Under the transitive hypothesis, we show the existence of sub-actions for H¨ older potentials also in the holonomic setting. We analyze then connections between calibrated sub-actions and the Ma˜ npotential. A representation formula for calibrated sub-actions is presented, which drives us naturally to a classification theorem for these sub-actions. We also investigate properties of the support of maximizing probabilities.

Functions for relative maximization

We introduce functions for relative maximization in a general context: the beta and alpha applications. After a systematic study of different kinds of regularities, we investigate how to approximate certain values of these functions using periodic orbits. We also show that the differential of an alpha application determines the asymptotic behavior of the optimal trajectories.

Zero-temperature phase diagram for double-well type potentials in the summable variation class

We study the zero-temperature limit of the Gibbs measures of a class of long-range potentials on a full shift of two symbols

THE EFFECTIVE POTENTIAL AND TRANSSHIPMENT IN THERMODYNAMIC FORMALISM AT ZERO TEMPERATURE

\{0,1\}

Aubry set for asymptotically sub-additive potentials

. These potentials were introduced by Walters as a natural space for the transfer operator. In our case, they are locally constant, Lipschitz continuous or, more generally, of summable variation. We assume there exists exactly two ground states: the fixed points

Average Sex Ratio and Population Maintenance Cost

0^\infty

Mañé Potential and Peierls Barrier

and

Separating Sub-actions

1^\infty

Further Properties of Sub-actions

. We fully characterize, in terms of the Peierls barrier between the two ground states, the zero-temperature phase diagram of such potentials, that is, the regions of convergence or divergence of the Gibbs measures as the temperature goes to zero.

Calibrated Sub-actions

For a topologically transitive subshift of finite type defined by a symmetric transition matrix, we introduce a temperature-based problem related to the usual thermodynamic formalism. This problem is described by an operator acting on Holder continuous observables which is actually superlinear with respect to the max-plus algebra. We thus show that, for each fixed absolute temperature, such an operator admits a unique eigenfunction and a unique eigenvalue. We also study the convergence as the temperature goes to zero and we relate the limit objects to an ergodic version of Kantorovich transshipment problem.