A monotone Sinai theorem

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.Comment: Published at http://dx.doi.org/10.1214/14-AOP968 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Chaos and Shadowing Around a Homoclinic Tube

Let $F$ be a $C^3$ diffeomorphism on a Banach space $B$. $F$ has a homoclinic tube asymptotic to an invariant manifold. Around the homoclinic tube, Bernoulli shift dynamics of submanifolds is established through shadowing lemma. This work removes an uncheckable condition of Silnikov [Equation (11), page 625 of L. P. Silnikov, Soviet Math. Dokl., vol.9, no.3, (1968), 624-628]. Also, the result of Silnikov does not imply Bernoulli shift dynamics of a single map, rather only provides a labeling of all invariant tubes around the homoclinic tube. The work of Silnikov was done in ${\mathbb R}^n$, and the current work is done in a Banach space.Comment: accepted, Abstract and Applied Analysi

Homoclinic Tubes and Chaos in Perturbed Sine-Gordon Equation

In an early work, Bernoulli shift dynamics of submanifolds was established in a neighborhood of a homoclinic tube. In this article, we will present a concrete example: sine-Gordon equation under a quasi-periodic perturbation

Aperiodic Sequences and Aperiodic Geodesics

We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition

Trajectory versus probability density entropy

We study the problem of entropy increase of the Bernoulli-shift map without recourse to the concept of trajectory and we discuss whether, and under which conditions if it does, the distribution density entropy coincides with the Kolmogorov-Sinai entropy, namely, with the trajectory entropy.Comment: 24 page

Solvable model for spatiotemporal chaos

We show that the dynamical behavior of a coupled map lattice where the individual maps are Bernoulli shift maps can be solved analytically for integer couplings. We calculate the invariant density of the system and show that it displays a nontrivial spatial behavior. We also introduce and calculate a generalized spatiotemporal correlation function