Multifractal formalism for Benedicks-Carleson quadratic maps

Abstract

For a positive measure set of nonuniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given observable and consider the associated {\it Birkhoff spectrum} which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to "see" sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle for empirical distributions, with Lebesgue as a reference measure.

25 pages, no figure, Ergodic Theory and Dynamical Systems, to appear

References in corpus (4)

• Bowen's equation in the non-uniform setting
• Large deviation principles for non-uniformly hyperbolic rational maps
• Large deviation principle for Benedicks-Carleson quadratic maps
• Dimension theory for multimodal maps

Cited by in corpus (7)

• Large Deviation Principle for $S$-unimodal maps with flat critical point
• Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems
• Lyapunov spectrum for Hénon-like maps at the first bifurcation
• The Historic Set of Ergodic Averages in Some Nonuniformly Hyperbolic Systems
• Large deviation principle in one-dimensional dynamics
• Multifractal Analysis of Ergodic Averages in Some Nonuniformly Hyperbolic Systems
• Birkhoff spectrum for Hénon-like maps at the first bifurcation