Alexandre Baraviera

Removing zero Lyapunov exponents

In an explicit family of partially hyperbolic diffeomorphisms of the torus T 3 , ShubandWilkinsonrecentlysucceededin perturbingthe Lyapunovexponentsofthe center direction. We present here a local version of their argument, allowing one to perturb the center Lyapunov exponents of any partially hyperbolic system, in any dimension and with arbitrary dimension of the center bundle.

A LARGE DEVIATION PRINCIPLE FOR THE EQUILIBRIUM STATES OF HÖLDER POTENTIALS: THE ZERO TEMPERATURE CASE

Consider a α-Holder function A : Σ → ℝ and assume that it admits a unique maximizing measure μmax. For each β, we denote μβ, the unique equilibrium measure associated to βA. We show that (μβ) satisfies a Large Deviation Principle, that is, for any cylinder C of Σ, \[ \lim_{\beta \to +\infty} \frac{1}{\beta} \log \mu_{\beta}(C)=-\inf_{x \in C} I(x) \] where \[ I(x)=\sum_{n\geq0}(V\circ\sigma-V-(A-m))\circ\sigma^n(x), \quad m=\int\!A\,d\mu_{\max} \] where V(x) is any strict subaction of A.

ON THE GENERAL ONE-DIMENSIONAL XY MODEL: POSITIVE AND ZERO TEMPERATURE, SELECTION AND NON-SELECTION

We consider (M, d) a connected and compact manifold and we denote by the Bernoulli space Mℤ. The analogous problem on the half-line ℕ is also considered. Let be an observable. Given a temperature T, we analyze the main properties of the Gibbs state . In order to do our analysis, we consider the Ruelle operator associated to , and we get in this procedure the main eigenfunction . Later, we analyze selection problems when the temperature goes to zero: (a) existence, or not, of the limit , a question about selection of subactions, and, (b) existence, or not, of the limit , a question about selection of measures. The existence of subactions and other properties of Ergodic Optimization are also considered. The case where the potential depends just on the coordinates (x0, x1) is carefully analyzed. We show, in this case, and under suitable hypotheses, a Large Deviation Principle, when T → 0, graph properties, etc. Finally, we will present in detail a result due to van Enter and Ruszel, where the authors show, for a particular example of potential A, that the selection of measure in this case, does not happen.

Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials

For the subshift of nite type = f0; 1; 2g N we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non-locally constant Holder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two dierent xed points and the potential is atter at one of these two xed points. We prove that there always is convergence but not necessarily to the Dirac measure at the point where the potential is the attest. This is contrary to what was expected in the light of the analogous problem in Aubry-Mather theory (1). This is also contrary to the nite range case where the equilibrium state converges to the equi-barycentre of the two Dirac measures. Moreover we emphasize the strange behavior of the Gibbs measure: the eigenmeasure selects one Dirac measure ( at the point where the potential is the attest) and the eigen- function selects the other one (at the point where the potential is the sharpest).

On the selection of subaction and measure for a subclass of potentials defined by P. Walters

Suppose

Emerging collective behavior in a simple artificial financial market

\sigma

Dynamics of a Public Investment Game: from Nearest-Neighbor Lattices to Small-World Networks

is the shift acting on Bernoulli space

A Thermodynamic Formalism for Density Matrices in Quantum Information

X=\{0,1\}^\mathbb{N}

On the Herman–Avila–Bochi formula for Lyapunov exponents of {\rm SL}(2,\mathbb{R}) -cocycles

, and, consider a fixed function

A Dynamical Point of View of Quantum Information: Discrete Wigner Measures

f:X \to \mathbb{R}