34 research outputs found
Thermodynamic formalism for Lorenz maps
For a 2-dimensional map representing an expanding geometric Lorenz at-
tractor we prove that the attractor is the closure of a union of as long as
possible unstable leaves with ending points. This allows to define the notion
of good measures, those giving full measure to the union of these open leaves.
Then, for any H\"older continuous potential we prove that there exists at most
one relative equilibrium state among the set of good measures. Condition
yielding existence are given.Comment: 36 page
Ergodic Formalism for topological Attractors and historic behavior
We introduce the concept of Baire Ergodicity and Ergodic Formalism. We use
them to study topological and statistical attractors, in particular to
establish the existence and finiteness of such attractors. We give applications
for maps of the interval, non uniformly expanding maps, partially hyperbolic
systems, strongly transitive dynamics and skew-products. In dynamical systems
with abundance of historic behavior (and this includes all systems with some
hyperbolicity, in particular, Axiom A systems), we cannot use an invariant
probability to control the asymptotic topological/statistical behavior of a
generic orbit. However, the results presented here can also be applied in this
context, contributing to the study of generic orbits of systems with abundance
of historic behavior.Comment: 37 pages, 4 figure
Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction
We consider a partially hyperbolic set on a Riemannian manifold whose
tangent space splits as , for which the
centre-unstable direction expands non-uniformly on some local unstable
disk. We show that under these assumptions induces a Gibbs-Markov
structure. Moreover, the decay of the return time function can be controlled in
terms of the time typical points need to achieve some uniform expanding
behavior in the centre-unstable direction. As an application of the main result
we obtain certain rates for decay of correlations, large deviations, an almost
sure invariance principle and the validity of the Central Limit Theorem.Comment: 23 page