Artur O. Lopes

Lyapunov minimizing measures for expanding maps of the circle

We consider the set of maps f 2 FC D( >C 1C of the circle which are covering maps of degree D, expanding, min x2S1f 0 .x/ > 1 and orientation preserving. We are interested in characterizing the set of such mapsf which admit a uniquef -invariant probability measure minimizing R lnf 0 d over all f -invariant probability measures. We show there exists a set GC FC, open and dense in the C 1C -topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, iff admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C 1C -topology of maps admitting a unique minimizing measure supported on a strictly

Entropy and large deviation

The author shows the existence of a deviation function for the maximal measure mu of a hyperbolic rational map of degree d. He relates several results of large deviation with the thermodynamic formalism of ergodic theory. The maximal measure plays a distinguished role among other invariant measures, because the stochastic process given by the rational map and the maximal measure will generate a free energy function, whose Legendre transform in the set of invariant measures will be log d minus the entropy in the sense of Shannon-Kolmogorov. This result is associated with the relation between pressure and free energy. A general description of the result is as follows. Consider mu to be the maximal entropy measure and v another invariant measure. The ergodic theorem claims that the mean of the sum of Dirac measures in the orbit of a mu -almost everywhere point z, will converge to mu . Given a convex neighbourhood G of v in the set of measures, the author estimates the deviations of the mean of the sum of Dirac measures in the orbit of a mu -almost everywhere point z, with respect to this neighbourhood G. If the neighbourhood G is very small and he considers large iterates, the exponential value of decreasing of the mu -measure of points whose mean orbit is in G approximately the entropy of v minus log d. In this way, he calculates the entropy of v as an information of large deviation related to the maximal measure mu . He applies this result, using a contraction principle, to measure the deviation of the Liapunov number of the maximal measure. The same proof presented in this paper also works (with minor modifications) for shifts of finite type in the lattice N.

The dimension spectrum of the maximal measure

A variety of complicated fractal objects and strange sets appears in nonlinear physics. In diffusion-limited aggregation, the probability of a random walker landing next to a given site of the aggregate is of interest. In percolation, the distribution of voltages across different elements in a random-resistor network (see [T. Halsey et al., Phys. Rev. A (3), 33 (1986), pp. 1141–1151]) may be of interest. These examples can be better analyzed by dividing certain objects in pieces labeled by indexes, but that leads to working with fractal sets and the notion of dimension [Halsey et al. (1986)].The dimension spectrum of a system has been introduced and measured experimentally, and a substantial literature in physics addresses this topic. In several important cases, rigorous proofs of the results presented in [Halsey et al. (1986)] have been established.Here, rigorous mathematical proofs of some results in this theory are given, specifically for the maximal entropy measure of a hyperbolic rational map in the ...

Open billiards: invariant and conditionally invariant probabilities on Cantor sets

Billiards are the simplest models for understanding the statistical theory of the dynamics of a gas in a closed compartment. We analyze the dynamics of a class of billiards (the open billiard on the plane) in terms of invariant and conditionally invariant probabilities. The dynamical system has a horseshoe structure. The stable and unstable manifolds are analytically described. The natural probability

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

\mu

On the Aubry-Mather theory for symbolic dynamics

is invariant and has support in a Cantor set. This probability is the conditional limit of a conditional probability

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

\mu _F

Entropy and variational principle for one-dimensional lattice systems with a general a priori probability: positive and zero temperature

that has a density with respect to the Lebesgue measure. A formula relating entropy, Lyapunov exponent, and Hausdorff dimension of a natural probability

Exact bounds for the polynomial decay of correlation, 1/f noise and the CLT for the equilibrium state of a non-Hölder potential

\mu

The Theta Group and the Continued Fraction Expansion with Even Partial Quotients

for the system is presented. The natural probability