Measures of Maximal Dimension for Hyperbolic Diffeomorphisms
نویسندگان
چکیده
We establish the existence of ergodic measures of maximal Hausdorff dimension for hyperbolic sets of surface diffeomorphisms. This is a dimension-theoretical version of the existence of ergodic measures of maximal entropy. The crucial difference is that while the entropy map is upper-semicontinuous, the map ν 7→ dimH ν is neither uppersemicontinuous nor lower-semicontinuous. This forces us to develop a new approach, which is based on the thermodynamic formalism. Remarkably, for a generic diffeomorphism with a hyperbolic set, there exists an ergodic measure of maximal Hausdorff dimension in a particular two-parameter family of equilibrium measures. We also provide versions of these results for conformal diffeomorphisms on higher-dimensional manifolds and conformal hyperbolic flows.
منابع مشابه
Intrinsic Ergodicity for Certain Nonhyperbolic Robustly Transitive Systems
We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have constant entropy and isomorphic unique measures of maximal entropy.
متن کاملVariational Properties of Multifractal Spectra
For hyperbolic diffeomorphisms, we describe the variational properties of the dimension spectrum of equilibrium measures on locally maximal hyperbolic sets, when the measure or the dynamical system are perturbed. We also obtain explicit expressions for the first derivative of the dimension spectra and the associated Legendre transforms. This allows us to establish a local version of multifracta...
متن کاملEquality of Pressures for Diffeomorphisms Preserving Hyperbolic Measures
For a uniformly hyperbolic diffeomorphism f , the induced volume deformation φu in the unstable subbundle over a compact f -invariant set significantly characterizes the geometry of the set as well as the dynamics in its neighborhood. Under the hypothesis of uniform hyperbolicity, and particularly for hyperbolic surface diffeomorphisms, a large number of dynamical quantifiers such as, for examp...
متن کاملOn pointwise dimension of non-hyperbolic measures
We construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we also show that such diffeomorphisms are typical in certain situations.
متن کاملDynamical Coherence of Partially Hyperbolic Diffeomorphisms of Tori Isotopic to Anosov
We show that partially hyperbolic diffeomorphisms of d-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anoso...
متن کامل