Symbolic dynamics for non-uniformly hyperbolic systems

Abstract

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

Chaos in spatially homogeneous Hořava-Lifshitz subcritical cosmologies

“…Our goal is to prove that the Kasner map is chaotic for v ∈ (0, 1/2), which thereby completes the program in [10] and rigorously shows that GR is a critical case associated with a bifurcation where chaos becomes generic. Even though irregular and non-hyperbolic maps impose difficulties in proving that a given map is chaotic, see [17], it is the fact that the Kasner map is multivalued that complicates the analysis.…”

Chaos in spatially homogeneous Hořava–Lifshitz subcritical cosmologies

“…GR is a specific model among Hor ˇava gravity where a qualitative change on the structure of solutions occurs. Even though irregular and non-hyperbolic maps impose difficulties in proving that a given map is chaotic, see [27], it is the fact that the Kasner map is multi-valued that complicates the analysis.…”

Bifurcations and Chaos in Hořava-Lifshitz Cosmology

“…Hence K is an example of a non-uniformly hyperbolic circle map, and it would be interesting to investigate its dynamical properties with recent mathematical methods developed in[45,12,54,22] and references therein.…”