45 research outputs found
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
Dimension and product structure of hyperbolic measures
We prove that every hyperbolic measure invariant under a C^{1+\alpha}
diffeomorphism of a smooth Riemannian manifold possesses asymptotically
``almost'' local product structure, i.e., its density can be approximated by
the product of the densities on stable and unstable manifolds up to small
exponentials. This has not been known even for measures supported on locally
maximal hyperbolic sets.
Using this property of hyperbolic measures we prove the long-standing
Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems: the
pointwise dimension of every hyperbolic measure invariant under a C^{1+\alpha}
diffeomorphism exists almost everywhere. This implies the crucial fact that
virtually all the characteristics of dimension type of the measure (including
the Hausdorff dimension, box dimension, and information dimension) coincide.
This provides the rigorous mathematical justification of the concept of fractal
dimension for hyperbolic measures.Comment: 29 pages, published versio
Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps
We prove the existence of SRB measures for diffeomorphisms where a positive
volume set of initial conditions satisfy an "effective hyperbolicity" condition
that guarantees certain recurrence conditions on the iterates of Lebesgue
measure. We give examples of systems that do not admit a dominated splitting
but can be shown to have SRB measures using our methods.Comment: 54 pages, significant restructuring of exposition and reorganization
of proofs for clarity. Main results remain the sam