262 research outputs found
Physical measures for infinite-modal maps
We analyze certain parametrized families of one-dimensional maps with
infinitely many critical points from the measure-theoretical point of view. We
prove that such families have absolutely continuous invariant probability
measures for a positive Lebesgue measure subset of parameters. Moreover we show
that both the densities of these measures and their entropy vary continuously
with the parameter. In addition we obtain exponential rate of mixing for these
measures and also that they satisfy the Central Limit Theorem.Comment: 41 pages, 3 figures, some proofs completely rewritten. Keywords: SRB
measures, absolutely continuous invariant measures, infinite-modal maps,
statistical stability, sub-exponential decay of correlations, central limit
theorem, continuous variation of entrop
Singular-hyperbolic attractors are chaotic
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is
chaotic, in two strong different senses. Firstly, the flow is expansive: if two
points remain close for all times, possibly with time reparametrization, then
their orbits coincide. Secondly, there exists a physical (or
Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin
covers a full Lebesgue (volume) measure subset of the topological basin of
attraction. Moreover this measure has absolutely continuous conditional
measures along the center-unstable direction, is a -Gibbs state and an
equilibrium state for the logarithm of the Jacobian of the time one map of the
flow along the strong-unstable direction. This extends to the class of
singular-hyperbolic attractors the main elements of the ergodic theory of
uniformly hyperbolic (or Axiom A) attractors for flows.Comment: 55 pages, extra figures (now a total of 16), major rearrangement of
sections and corrected proofs, improved introductio
Entropy-expansiveness for partially hyperbolic diffeomorphisms
We show that diffeomorphisms with a dominated splitting of the form
, where is a nonhyperbolic central bundle that
splits in a dominated way into 1-dimensional subbundles, are entropy-expansive.
In particular, they have a principal symbolic extension and equilibrium states.Comment: 15 pages, 1 figur
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