arXiv subjects

Hiroki Takahasi

Publications and source records attributed to Hiroki Takahasi.

At least 55 records · Page 3Linked to original sources

arXiv2019

Large Deviation Principles for countable Markov shifts

We establish the large deviation principle for a topological Markov shift over infinite alphabet which satisfies strong combinatorial assumptions called ``finite irreducibility'' or ``finite primitiveness''. More precisely, we assume the existence of a Gibbs state for a potential $ϕ$ in the sense of Bowen, and prove the level-2 Large Deviation Principles for the distribution of empirical means under the Gibbs state, as well as that of weighted periodic points and iterated pre-images. The rate function is written with the pressure and the free energy associated with the potential $ϕ$.

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arXiv2018

Large Deviation Principle for arithmetic functions in continued fraction expansion

Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity.

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arXiv2018

Boundary of the horseshoe locus for the Hénon family

The purpose of this article is to investigate geometric properties of the parameter locus of the Hénon family where the uniform hyperbolicity of a horseshoe breaks down. As an application, we obtain a variational characterization of equilibrium measures "at temperature zero" for the corresponding non-uniformly hyperbolic Hénon maps. The method of the proof also yields that the boundary of the hyperbolic horseshoe locus in the parameter space consists of two monotone pieces, which confirms a conjecture in [AI]. The proofs of these results are based on the machinery developed in [AI] which employs the complexification of both the dynamical and the parameter spaces of the Hénon family together with computer assistance.

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arXiv2017

Large Deviation Principle for $S$-unimodal maps with flat critical point

We study a topologically exact, negative Schwarzian unimodal map whose critical point is non-recurrent and flat. Assuming the critical order is either logarithmic or polynomial, we establish the Large Deviation Principle and give a partial description of the zeros of the corresponding rate functions. We apply our main results to a certain parametrized family of unimodal maps in the same topological conjugacy class, and give a complete description of the zeros of the rate functions. We observe a qualitative change at a transition parameter, and show that the sets of zeros depend continuously on the parameter even at the transition.

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arXiv2017

Equilibrium states at freezing phase transition in unimodal maps with flat critical point

An $S$-unimodal map $f$ with flat critical point satisfying the Misiurewicz condition displays a freezing phase transition in positive spectrum. We analyze statistical properties of the equilibrium state $μ_t$ for the potential $-t\log|Df|$, as well as how the phase transition slows down the rate of decay of correlations. We show that $μ_t$ has exponential decay of correlations for all inverse temperature $t$ contained in the positive entropy phase $(t^-,t^+)$. If the critical point is not too flat, then the freezing point $t^+$ is equal to $1$, and the absolutely continuous invariant probability measure (acip for short) is the unique equilibrium state at the transition. We exhibit a case in which the acip has sub-exponential decay of correlations and $μ_t$ converges weakly to the acip as $t\nearrow t^+$.

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arXiv2017

Lyapunov optimization for non-generic one-dimensional expanding Markov maps

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^1$ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

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arXiv2016

Removal of phase transition of the Chebyshev quadratic and thermodynamics of Hénon-like maps near the first bifurcation

We treat a problem at the interface of dynamical systems and equilibrium statistical physics. It is well-known that the geometric pressure function $$t\in\mathbb R\mapsto \sup_μ\left\{h_μ(T_2)-t\int\log |dT_2(x)|dμ(x)\right\}$$ of the Chebyshev quadratic map $T_2(x)=1-2x^2$ $(x\in\mathbb R)$ is not differentiable at $t=-1$. We show that this phase transition can be "removed", by an arbitrarily small singular perturbation of the map $T_2$ into Hénon-like diffeomorphisms. A proof of this result relies on an elaboration of the well-known inducing techniques adapted to Hénon-like dynamics near the first bifurcation.

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arXiv2015

Lyapunov spectrum for Hénon-like maps at the first bifurcation

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e., decompose the set of non wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries a full Hausdorff dimension.

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arXiv2015

Birkhoff spectrum for Hénon-like maps at the first bifurcation

We effect a multifractal analysis for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We decompose the set of non wandering points on the unstable manifold into level sets of Birkhoff averages of continuous functions, and derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures.

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arXiv2014

Equilibrium measures at temperature zero for Hénon-like maps at the first bifurcation

We develop a thermodynamic formalism for a strongly dissipative Hénon-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 $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as $t\to+\infty$ minimizes the unstable Lyapunov exponent. We also show that the equilibrium measures converge as $t\to-\infty$ to a Dirac measure which maximizes the unstable Lyapunov exponent.

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arXiv2014

Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

For strongly dissipative Hénon maps at the first bifurcation where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e., prove the existence and uniqueness of an invariant probability measure which maximizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $t\in\mathbb R$ is in a certain large interval and $J^u$ is the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

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arXiv2014

Prevalence of non-uniform hyperbolicity at the first bifurcation of Hénon-like families

We consider the dynamics of strongly dissipative Hénon-like maps in the plane, around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of homoclinic or heteroclinic tangencies inside the limit set. In [Takahasi H.: Commun. Math. Phys. {\bf 312} 37-85 (2012)], it was proved that $a^*$ is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under forward iteration. For these parameters, we show that all Lyapunov exponents of all invariant ergodic Borel probability measures are uniformly bounded away from zero, uniformly over all the parameters.

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arXiv2013

Equilibrium measures for the Hénon map at the first bifurcation

We study the dynamics of strongly dissipative Hénon 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 $-t\log J^u$, where $t\in\mathbb R$ is in a certain interval of the form $(-\infty,t_0)$, $t_0>1$ and $J^u$ denotes the Jacobian in the unstable direction.

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arXiv2012

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.

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arXiv2012

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.

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arXiv2011

Nonuniformly expanding 1d maps with logarithmic singularities

For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behavior. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.

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arXiv2010

Critical points for surface maps and the Benedicks-Carleson theorem

We give an alternative proof of the Benedicks-Carleson theorem on the existence of strange attractors in Hénon-like families in the plane. To bypass a huge inductive argument, we introduce an induction-free explicit definition of dynamically critical points. The argument is sufficiently general and in particular applies to the case of non-invertible maps as well. It naturally raises the question of an intrinsic characterization of dynamically critical points for dissipative surface maps.

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arXiv2010

Prevalent dynamics at the first bifurcation of Henon-like families

We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that $a^*$ is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that $a^*$ corresponds to "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusion argument, we construct a set of "good parameters" having $a^*$ as a full density point. Adapting Benedicks & Viana's volume control argument on the other, we analyze Lebesgue typical dynamics corresponding to these good parameters.

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