A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.Comment: 35 page

Ergodicity of polygonal slap maps

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.Comment: 17 pages, 6 figure

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

We study polygonal billiards with reflection laws contracting the reflected angle towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic SRB measures whose basins cover a set of full Lebesgue measure.Comment: 26 pages, 2 figure

Asymptotic stability of the optimal filter for random chaotic maps

The asymptotic stability of the optimal filtering process in discrete time is revisited. The filtering process is the conditional probability of the state of a Markov process, called the signal process, given a series of observations. Asymptotic stability means that the distance between the true filtering process and a wrongly initialised filter converges to zero as time progresses. In the present setting, the signal process arises through iterating an i.i.d. sequence of uniformly expanding random maps. It is showed that for such a signal, the asymptotic stability is exponential provided that its initial conditions are sufficiently smooth. Similar to previous work on this problem, Hilbert’s projective metric on cones is employed as well as certain mixing properties of the signal, albeit with important differences. Mixing and ultimately filter stability in the present situation are due to the expanding dynamics rather than the stochasticity of the signal process. In fact, the conditions even permit iterations of a fixed (nonrandom) expanding map

Bernoulli Elliptical Stadia

Let Q_{a,h} be a convex region of the plane whose boundary consists of two semiellipses joined by two (straight) lines parallel to the major axis of the semiellipses (elliptical stadium). The axes of the semiellipses have length 2 and 2a,a>1, and the lines have length 2h. For and we give a complete proof of the following result: the billiard map in the elliptical stadium Q_{a,h} is ergodic, K-mixing and Bernoulli with respect to the natural billiard measure

On the Bernoulli Property of Planar Hyperbolic Billiards

We consider billiards in non-polygonal domains of the plane with boundaryconsisting of curves of three different types: straight segments, strictly convex inwardcurves and strictly convex outward curves of a special kind. The billiard map for thesedomains is known to have non-vanishing Lyapunov exponents a.e. provided that thedistance between the curved components of the boundary is sufficiently large, and the setof orbits having collisions only with the flat part of the boundary has zero measure. Undera few additional conditions, we prove that there exists a full measure set of the billiardphase space such that each of its points has a neighborhood contained up to a zero measureset in one Bernoulli component of the billiard map. Using this result, we show that thereexists a large class of planar hyperbolic billiards that have the Bernoulli property. This class includes the billiards in convex domains bounded by straight segments and strictlyconvex inward arcs constructed by Donnay