Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

Abstract

We consider maps preserving a foliation which is uniformly contracting and a onedimensional 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. -hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm law. V.A. and M.J.P. were partially supported by CNPq, PRONEX-Dyn.Syst., FAPERJ, Balzan Research Project of J.Palis .

Erratum to: Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

“…In this note, we explain how the construction of the global Poincaré map R can be adjusted, in a similar way to the one originally presented in [2], to obtain [1, Corollary 1] (where the n-th return map F considered in the statement is replaced by the map R defined here) and [1,Corollary 2], which are the main results of the second part of [1]. …”

“…Moreover, the application to the results given in [1,Sect. 7] has the following problem: The almost everywhere existence of local dimension for the invariant measure might fail.…”

“…We now show how to change [1,Sect. 7] adapting it to the return map which is considered here, in a way that [1, Corollary 2] is proved in the same form as it is stated.…”

“…We separate our corrections into three parts, which apply for the definition of R here presented in (1.1) and lead to the same goal of the above-mentioned Remark 17 in [1].…”

“…The modification was done to provide injectivity. Injectivity of the global Poincaré map is important for the arguments in [1] where the physical measure is shown to be exact dimensional.…”