arXiv subjects
Publications and source records attributed to Sebastien Gouezel.
We prove that solutions $u$ of the equation $f=u-u\circ T$ are automatically Holder continuous when $f$ is Holder continuous, and $T$ is non uniformly expanding and Markov. This result applies in particular to Young towers and to intermittent maps.
We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a $\sqrt{n\log n}$ normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.
We prove that, in Young towers with sufficiently small tails, the speed in the central limit theorem is O(1/\sqrt{n}), and the local limit theorem holds. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.
We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O(e^{-c \sqrt{n}})$.
We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a stable law, depending on the average of the observable along the neutral curve.
In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form $x+x^{1+α}$, for $α\in (0,1)$. In particular, for $α>1/2$, we show that the Birkhoff sums of a Hölder observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in noncommutative Banach algebras.
We generalize a method developed by Sarig to obtain polynomial lower bounds for correlation functions for maps with a countable Markov partition. A consequence is that LS Young's estimates on towers are always optimal. Moreover, we show that, for functions with zero average, the decay rate is better, gaining a factor 1/n. This implies a Central Limit Theorem in contexts where it was not expected, e.g. x+Cx^(1+α) with 1/2 < α< 1. The method is based on a general result on renewal sequences of operator, and gives an asymptotic estimate up to any precision of such operators.