Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations

Ergodic properties of infinite extensions of area-preserving flows

We consider volume-preserving flows $(\Phi^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(\Phi^f_t)_{t\in\mathbb{R}}$ has the form $\Phi^f_t(x,y)=(\phi_tx,y+\int_0^t f(\phi_sx)ds)$, where $(\phi_t)_{t\in\mathbb{R}}$ is a locally Hamiltonian flow of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on $S$. We investigate ergodic properties of these infinite measure-preserving flows and prove that if $f$ belongs to a space of finite codimension in $\mathscr{C}^{2+\epsilon}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(\phi_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, then $(\Phi^f_t)_{t\in\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(\Phi^0_t)_{t\in\mathbb{R}}$. The proof of this result exploits the reduction of $(\Phi^f_t)_{t\in\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(\phi_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.Comment: 57 pages, 4 picture

A temporal Central Limit Theorem for real-valued cocycles over rotations

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $\alpha$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $\beta$. When $\alpha$ is badly approximable and $\beta$ is badly approximable with respect to $\alpha$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $\alpha$ is quadratic irrational, $\beta$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of $\beta$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.Comment: a few typos corrected, 28 pages, 4 figure