We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.
On the Finiteness of Attractors for One-Dimensional Maps with Discontinuities
Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps
Stable sets, hyperbolicity and dimension
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