Dimensional properties of dynamical systems with discrete and continuous time.

摘要

We develop the multifractal analysis of conformal axiom A flows. This includes the study of the Hausdorff dimension of basic sets of the flow, the description of the dimension spectra for pointwise dimension and for Lyapunov exponents and the multifractal decomposition associated with these spectra. The main tool of study is the thermodynamic formalism for hyperbolic flows by Bowen and Ruelle. Examples include suspensions over axiom A conformal diffeomorphisms, Anosov flows, and in particular, geodesic flows on compact smooth surfaces of negative curvature.;We also construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we show that such diffeomorphisms are typical in certain situations.