157 research outputs found
Multifractal formalism for Benedicks-Carleson quadratic maps
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.Comment: 25 pages, no figure, Ergodic Theory and Dynamical Systems, to appea
Quenched limit theorems for random U(1) extensions of expanding maps
The Lyapunov spectra of random U(1) extensions of expanding maps on the torus
were investigated in our previous work [NW2015]. Using the result, we extend
the recent spectral approach for quenched limit theorems for expanding maps
[DFGV2018] and hyperbolic maps [DFGV2019] to our partially hyperbolic dynamics.
Quenched central limit theorems, large deviations principles and local central
limit theorems for random U(1) extensions of expanding maps on the torus are
proved via corresponding theorems for abstract random dynamical systems.Comment: 39 page
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