Periodic billiard orbits in right triangle

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface

Residual generic ergodicity of periodic group extensions over translation surfaces

Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group extension produced using a technique referred to as \emph{cuts}

Entropy and Complexity of Polygonal Billiards with Spy Mirrors

We prove that a polygonal billiard with one-sided mirrors has zero topological entropy. In certain cases we show sub exponential and for other polynomial estimates on the complexity

Pseudo-physical measures for typical continuous maps of the interval

We study the measure theoretic properties of typical C 0 maps of the interval. We prove that any ergodic measure is pseudo-physical, and conversely, any pseudo-physical measure is in the closure of the ergodic measures, as well as in the closure of the atomic measures. We show that the set of pseudo-physical measures is meager in the space of all invariant measures. Finally, we study the entropy function. We construct pseudo-physical measures with infinite entropy. We also prove that, for each m $\ge$ 1, there exists infinitely many pseudo-physical measures with entropy log m, and deduce that the entropy function is neither upper semi-continuous nor lower semi-continuous

Infinite-horizon Lorentz tubes and gases: recurrence and ergodic properties

We construct classes of two-dimensional aperiodic Lorentz systems that have infinite horizon and are 'chaotic', in the sense that they are (Poincar\'e) recurrent, uniformly hyperbolic, ergodic, and the first-return map to any scatterer is $K$-mixing. In the case of the Lorentz tubes (i.e., Lorentz gases in a strip), we define general measured families of systems (\emph{ensembles}) for which the above properties occur with probability 1. In the case of the Lorentz gases in the plane, we define families, endowed with a natural metric, within which the set of all chaotic dynamical systems is uncountable and dense.Comment: Final version, to appear in Physica D (2011

The role of continuity and expansiveness on leo and periodic specification properties

In this short note we prove that a continuous map which is locally eventually onto and is expansive satisfies the periodic specification property. We also discuss the role of continuity as a key condition in the previous characterization. We include several examples to illustrate the relation between these concepts.Comment: Theorem 1 needed an extra hypothesis, example 10 shows the necessity of this hypothesi

Coding discretizations of continuous functions

We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical continuous function when the diameter of the discretization tends to zero. Our main result is that any finite word appears on a subsequence discretization with any desired limit frequency