1,249 research outputs found
Absolute Continuity Theorem for Random Dynamical Systems on
In this article we provide a proof of the so called absolute continuity
theorem for random dynamical systems on which have an invariant
probability measure. First we present the construction of local stable
manifolds in this case. Then the absolute continuity theorem basically states
that for any two transversal manifolds to the family of local stable manifolds
the induced Lebesgue measures on these transversal manifolds are absolutely
continuous under the map that transports every point on the first manifold
along the local stable manifold to the second manifold, the so-called
Poincar\'e map or holonomy map. In contrast to known results, we have to deal
with the non-compactness of the state space and the randomness of the random
dynamical system.Comment: 46 page
The thermodynamic approach to multifractal analysis
Most results in multifractal analysis are obtained using either a
thermodynamic approach based on existence and uniqueness of equilibrium states
or a saturation approach based on some version of the specification property. A
general framework incorporating the most important multifractal spectra was
introduced by Barreira and Saussol, who used the thermodynamic approach to
establish the multifractal formalism in the uniformly hyperbolic setting,
unifying many existing results. We extend this framework to apply to a broad
class of non-uniformly hyperbolic systems, including examples with phase
transitions. In the process, we compare this thermodynamic approach with the
saturation approach and give a survey of many of the multifractal results in
the literature.Comment: 51 pages, minor corrections, added formal statements of new results
to "applications" sectio
Equilibrium Measures for Maps with Inducing Schemes
We introduce a class of continuous maps f of a compact metric space I
admitting inducing schemes and describe the tower constructions associated with
them. We then establish a thermodynamical formalism, i.e., describe a class of
real-valued potential functions \phi on I which admit unique equilibrium
measures \mu_\phi minimizing the free energy for a certain class of measures.
We also describe ergodic properties of equilibrium measures including decay of
correlation and the Central Limit Theorem. Our results apply in particular to
some one-dimensional unimodal and multimodal maps as well as to
multidimensional nonuniformly hyperbolic maps admitting Young's tower. Examples
of potential functions to which our theory applies include \phi_t=-t\log|df|
with t\in(t_0, t_1) for some t_0<1<t_1. In the particular case of S-unimodal
maps we show that one can choose t_0<0 and that the class of measures under
consideration comprises all invariant Borel probability measures. Thus our
results establish existence and uniqueness of both the measure of maximal
entropy (by a different method than Hofbauer) and the absolutely continuous
invariant measure extending results by Bruin and Keller for the parameters
under consideration
Bowen's equation in the non-uniform setting
We show that Bowen's equation, which characterises the Hausdorff dimension of
certain sets in terms of the topological pressure of an expanding conformal
map, applies in greater generality than has been heretofore established. In
particular, we consider an arbitrary subset Z of a compact metric space and
require only that the lower Lyapunov exponents be positive on Z, together with
a tempered contraction condition. Among other things, this allows us to compute
the dimension spectrum for Lyapunov exponents for maps with parabolic periodic
points, and to relate the Hausdorff dimension to the topological entropy for
arbitrary subsets of symbolic space with the appropriate metric.Comment: 23 pages, 1 figure: v2 has expanded introduction; "bounded"
contraction replaced with "tempered"; Section 4, Proposition 5.1 added; proof
of Lemma 6.2 correcte
- …