Paulo Varandas

Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle-Perron-Frobenius operator acting on the space of Holder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

Rapid Mixing for the Lorenz Attractor and Statistical Limit Laws for Their Time-1 Maps

We prove that every geometric Lorenz attractor satisfying a strong dissipativity condition has superpolynomial decay of correlations with respect to the unique Sinai–Ruelle–Bowen measure. Moreover, we prove the central limit theorem and almost sure invariance principle for the time-1 map of the flow of such attractors. In particular, our results apply to the classical Lorenz attractor.

Robust Exponential Decay of Correlations for Singular-Flows

We construct open sets of Ck (k ≥ 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.

Specification and thermodynamical properties of semigroup actions

In the present paper, we study the thermodynamical properties of finitely generated continuous subgroup actions. We propose a notion of topological entropy and pressure functions that do not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps, obtaining a partial extension of the results obtained by Ruelle for ℤd-actions [D. Ruelle, Trans. Am. Math. Soc., 187, 237–251 (1973)]. The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps are also discussed.

Entropy and Poincaré recurrence from a geometrical viewpoint

We study Poincare recurrence from a purely geometrical viewpoint. In (Downarowicz and Weiss 2004 Illinois J. Math. 48 59–69) it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and to prove that minimal return times to dynamical balls grow linearly with respect to its length. Some relations using weighted versions of recurrence times are also obtained for equilibrium states. Then we establish some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures.

Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if

Open sets of Axiom A flows with exponentially mixing attractors

f

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

has an expanding repeller and

Weak specification properties and large deviations for non-additive potentials

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Semigroup Actions of Expanding Maps

is a Holder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval