Hiroki Takahasi

Large Deviation Principle for Benedicks-Carleson Quadratic Maps

Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks & Carleson to build induced Markov maps of arbitrarily small scale and associated towers, to which the absolutely continuous measures can be lifted. These various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle, for the absolutely continuous measures. This result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.

Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

We formulate and prove a Jakobson?Benedicks?Carleson-type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family fa(x) = x2 ? a which have an absolutely continuous invariant probability measure is at least 10?5000.

Equilibrium measures for the Hénon map at the first bifurcation

We study the dynamics of strongly dissipative Henon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove the existence of an equilibrium measure which minimizes the free energy associated with the non-continuous potential −t logJu, where is in a certain interval of the form (−∞, t0), t0 > 0 and Ju denotes the Jacobian in the unstable direction.

Multifractal formalism for Benedicks–Carleson quadratic maps

For a positive measure set of non-uniformly 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 continuous function and consider the associated 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 of empirical distributions, with Lebesgue as a reference measure.

Equilibrium Measures at Temperature Zero for Hénon-Like Maps at the First Bifurcation

We develop a thermodynamic formalism for a strongly dissipative Henon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any

Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities

t\in\mathbb R

Nonuniformly expanding 1D maps with logarithmic singularities

we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a noncontinuous geometric potential

Removal of Phase Transition in the Chebyshev Quadratic and Thermodynamics for Hénon-Like Maps Near the First Bifurcation

-t\log J^u

Birkhoff spectrum for Hénon-like maps at the first bifurcation

, where

Magnetism of (Fe, Co)–Se System with a NiAs-type Structure

J^u