Vitor Araujo

Exponential Decay of Correlations for Nonuniformly Hyperbolic Flows with a \({{C^{1+\alpha}}}\) Stable Foliation, Including the Classical Lorenz Attractor

We prove exponential decay of correlations for a class of

Robust Exponential Decay of Correlations for Singular-Flows

{C^{1+alpha}}

Open sets of Axiom A flows with exponentially mixing attractors

C1+α uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.

Infinitesimal Lyapunov functions for singular flows

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.

Dominated splitting for exterior powers and singular hyperbolicity

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.

Dominated splittings for flows with singularities

We consider maps preserving a foliation which is uniformly contracting and a one-dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for suitable Poincaré maps of a large class of singular hyperbolic flows. From this we deduce a logarithm law for these flows.

Finiteness results for subgroups of finite extensions

For any dimension

Absolutely continuous invariant measures for random non-uniformly expanding maps

dgeq 3