Cauchy distributions for the integrable standard map

We consider the integrable (zero perturbation) two--dimensional standard map, in light of current developments on ergodic sums of irrational rotations, and recent numerical evidence that it might possess non-trivial q-Gaussian statistics. Using both classical and recent results, we show that the phase average of the sum of centered positions of an orbit, for long times and after normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for almost all individual orbits such a sum does not obey any distribution at all. We discuss the question of existence of distributions for KAM tori.Comment: 6 pages, 2 figure

No temporal distributional limit theorem for a.e. irrational translation

Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus such that the set of angles alpha for which the Birkhoff sums of f with respect to the irrational translation by alpha satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set of angles is equal to zero

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

On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of $S_{N}(f, \alpha) := \sum_{n=1}^N f(n\alpha)$ for almost every $\alpha$ where $N \in \mathbb{N}$ is ranging over a long interval and $f$ is a $1$-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for $N$ in a set of positive upper density, the order of $S_{N}(f, \alpha)$ is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy-Koksma inequality for such $f$, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel-Bernstein Theorem that allows very general modularity conditions.Comment: 34 pages, comments are welcom

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser–Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry. This year special emphasis where laid on symplectic methods with applications to dynamics. The workshop was dedicated to the memory of John Mather, Jean-Christophe Yoccoz and Krzysztof Wysocki

On Roth type conditions, duality and central Birkhoff sums for i.e.m

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.Comment: 45 pages, 5 figure