133 research outputs found
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
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 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
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
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 -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
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