arXiv subjects

Hiroki Takahasi

Publications and source records attributed to Hiroki Takahasi.

At least 37 records · Page 2Linked to original sources

arXiv2023

Restricted slowly growing digits for infinite iterated function systems

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $Λ(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in Λ(\mathbf{f}): a_n(x)\in D \text{ for all }n\in\mathbb N \text{ and }\lim_{n\to\infty} a_n=\infty\}. \] For a function $φ:\mathbb{N}\to [\min D, \infty)$ such that $φ(n)\to\infty$ as $n\to\infty$, we compute the Hausdorff dimension of the set $$ S(\mathbf{f},D,φ) = \left\{ x\in \E(\mathbf{f},D) : a_n(x)\leq φ(n) \text{ for all } n\in\mathbb N \right\}. $$ We prove that the Hausdorff dimension stays the same no matter how slowly the function $φ$ grows. One of the consequences of our result is the recent work of Takahasi (2023), which only dealt with regular continued fraction expansions. We further extend our result to slowly growing products of (not necessarily consecutive) digits.

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arXiv2023

Exponential mixing for heterochaos baker maps and the Dyck system

We investigate mixing properties of piecewise affine non-Markovian maps acting on $[0,1]^2$ or $[0,1]^3$ and preserving the Lebesgue measure, which are natural generalizations of the {\it heterochaos baker maps} introduced in [Y. Saiki, H. Takahasi, J. A. Yorke. Nonlinearity 34 (2021) 5744-5761]. These maps are skew products over uniformly expanding or hyperbolic bases, and the fiber direction is a center in which both contracting and expanding behaviors coexist. We prove that these maps are mixing of all orders. For maps with a mostly expanding or contracting center, we establish exponential mixing for Hölder continuous functions. Using this result, for the Dyck system originating in the theory of formal languages, we establish exponential mixing with respect to its two coexisting ergodic measures of maximal entropy.

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arXiv2023

Level-2 large deviation principle for countable Markov shifts without Gibbs states

We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen's Gibbs state. To deal with non-compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations. Under this condition we establish the level-2 Large Deviation Principle. We apply our results to the continued fraction expansion of real numbers in $[0,1)$ generated by the Rényi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the Rényi map.

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arXiv2023

Large deviations of homological growth rates for hyperbolic surfaces

We perform a large deviations analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. For surfaces uniformized by a wide class of Fuchsian groups of the first kind, we prove the existence of the rate function which estimates exponential probabilities with which the homological growth rates stay away from the mean value. The rate function is given in terms of the multifractal dimension spectrum described in our earlier result [arXiv:2204.08907]. We also establish an Erdős-Rényi law, and refined large deviations upper bounds.

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arXiv2023

Multifractal analysis of homological growth rates for hyperbolic surfaces

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

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arXiv2023

Hausdorff dimension of Cantor intersections and robust heterodimensional cycles for heterochaos horseshoe maps

As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly $2$ whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. As a corollary we detect $C^2$-robust heterodimensional cycles. Our proof employs the theory of normally hyperbolic invariant manifolds and the thicknesses of Cantor sets.

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arXiv2023

The twisted baker map

As a model to provide a hands-on, elementary understanding of "vortex dynamics", we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalized Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.

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arXiv2022

Piecewise-linear maps with heterogeneous chaos

Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two more explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.

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arXiv2022

Heterochaos baker maps and the Dyck system: maximal entropy measures and a mechanism for the breakdown of entropy approachability

We introduce two parametrized families of piecewise affine maps on $[0,1]^2$ and $[0,1]^3$, as generalizations of the heterochaos baker maps which were introduced and investigated in [Y. Saiki, H. Takahasi, J. A. Yorke, Nonlinearity, 34 (2021), 5744--5761] as minimal models of the unstable dimension variability in multidimensional dynamical systems. We show that natural coding spaces of these maps coincide with the Dyck system that has come from the theory of languages. Based on this coincidence, we start to develop a complementary analysis on their invariant measures. As a first attempt, we show the existence of two ergodic measures of maximal entropy for the generalized heterochaos baker maps. We also clarify a mechanism for the breakdown of entropy approachability.

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arXiv2022

Distribution of cycles for one-dimensional random dynamical systems

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of "random cycles", with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms. We also prove another almost-sure convergence theorem, as well as an averaged (annealed) theorem that is related to semigroup actions. We apply our results to the random $β$-expansion of real numbers, and obtain almost-sure convergences of average digital quantities in random $β$-expansions of random cycles that do not follow from the application of the ergodic theorems of Birkhoff or Kakutani. Our main results are applicable to random dynamical systems generated by finitely many maps with common neutral fixed points.

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arXiv2021

Hausdorff dimension of sets with restricted, slowly growing partial quotients

I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\geq f(n)$, where $B$ is an infinite subset of $\mathbb N$ and $f$ is a rapidly growing function with $n$. We show that, for an arbitrary $B$ and an arbitrary $f$ with values in $[\min B,\infty)$ and tending to infinity, the set of irrational numbers in $(0,1)$ such that \[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\] is of Hausdorff dimension $τ(B)/2,$ where $τ(B)$ is the exponent of convergence of $B$.

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arXiv2021

Statistical properties of periodic points for infinitely renormalizable unimodal maps

For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $φ$ which is a continuous or a geometric potential $φ=-β\log|f'|$ ($β\in\mathbb R$), we establish the level-2 Large Deviation Principle for weighted periodic points. From this, we deduce that all weighted periodic points equidistribute with respect to equilibrium states for the potential $φ$. In particular, it follows that all periodic points are equidistributed with respect to measures of maximal entropy, and all periodic points weighted with their Lyapunov exponents are equidistributed with respect to the post-critical measure supported on the attracting Cantor set.

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arXiv2020

Precise asymptotics on the Birkhoff sums for dynamical systems

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in $(0,1)$.

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arXiv2020

Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches

For a Markov map of an interval or the circle with countably many branches and finitely many neutral periodic points, we establish conditional variational formulas for the mixed multifractal spectra of Birkhoff averages of countably many observables, in terms of the Hausdorff dimension of invariant probability measures. Using our results, we are able to exhibit new fractal-geometric results for backward continued fraction expansions of real numbers, answering in particular a question of Pollicott. Moreover, we establish formulas for multi-cusp winding spectra for the Bowen-Series maps associated with finitely generated free Fuchsian groups with parabolic elements.

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arXiv2020

Uniqueness of minimizer for countable Markov shifts and equidistribution of periodic points

For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowen's Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that all periodic points weighted with the potential equidistribute with respect to the Gibbs-equilibrium state as the periods tend to infinity. Applications are given to the Gauss map, and the Bowen-Series map associated with a finitely generated free Fuchsian group with parabolic elements.

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arXiv2019

Large deviation principle for harmonic/geometric/arithmetic mean of digits in backward continued fraction expansion

We establish the (level-1) large deviation principles for three kinds of means associated with the backward continued fraction expansion. We show that: for the harmonic and geometric means, the rate functions vanish exactly at one point; for the arithmetic mean, it is completely degenerate, vanishing at every point in its effective domain. Our method of proof employs the thermodynamic formalism for finite Markov shifts, and a multifractal analysis for the Rényi map generating the backward continued fraction digits. We completely determine the class of unbounded arithmetic functions for which the rate functions vanish at every point in unbounded intervals.

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arXiv2019

Large deviation principle in one-dimensional dynamics

We study the dynamics of smooth interval maps with non-flat critical points. For every such a map that is topologically exact, we establish the full (level-2) Large Deviation Principle for empirical means. In particular, the Large Deviation Principle holds for every non\nobreakdash-renormalizable quadratic map. This includes the maps without physical measure found by Hofbauer and Keller, and challenges the widely-shared view of the Large Deviation Principle as a refinement of laws of large numbers.

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arXiv2019

Large deviations for denominators of continued fractions

We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{nπ^2}{12\log2}$. The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation.

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