arXiv subjects

Yuya Arima

Publications and source records attributed to Yuya Arima.

8 recordsLinked to original sources

arXiv2026

Existence of the Lyapunov exponent for $S$-unimodal maps

In this paper, we show that for any $S$-unimodal map $T$ on $[0,1]$ with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant $\lambda_T\in\mathbb{R}$. Moreover, $\lambda_T=0$ if and only if $T$ admits neither an absolutely continuous $T$-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an $S$-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every $x\in [0,1]$ the Lyapunov exponent along the orbit of $x$ exists and is equal to $0$. A key ingredient is the following result of independent interest. If an $S$-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every $x\in [0,1]$ the lower Lyapunov exponent along the orbit of $x$ is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.

math.DS↗

arXiv2026

Linear response asymmetry between SRB and physical measures for families of intermittent maps with a transition point

We study linear response for families of intermittent maps whose SRB measure undergoes a transition from finite to infinite total mass at a critical parameter value. Our results reveal the following fundamental asymmetry arising from this transition. Smooth parameter dependence of the SRB measure implies continuity of the physical measure at the transition point, while simultaneously precluding its differentiability there. In particular, although the physical measure varies continuously with respect to the parameter at the transition, it fails to admit a linear response for a large class of potentials in the usual sense. We derive an explicit one-sided derivative formula describing this singular behavior and thereby give a quantitative characterization of how statistical properties degenerate as the total mass of the SRB measure diverges. The key ingredient in the proof of our main theorem is a new method that relates the parameter dependence of physical measures near the transition point to the behavior of the Riemann zeta function near its pole at 1.

math.DS↗

arXiv2025

Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding R\'{e}nyi interval maps with countably many branches

In this paper, we study the multifractal spectrum of Birkhoff averages for non-uniformly expanding R\'{e}nyi interval maps with countably many branches. Our main theorem substantially strengthens conditional variational formulas established by Jaerisch and Takahasi. Furthermore, our results enable a detailed analysis of Khinchin exponents and arithmetic means of backward continued fraction expansions in terms of the Hausdorff dimension. We also give a positive answer to the conjecture of Jaerisch and Takahasi. In addition, we develop the thermodynamic formalism for non-uniformly expanding R\'{e}nyi interval maps with countably many branches.

math.DS↗

arXiv2025

Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding interval maps with finitely many branches

In this paper, we perform a multifractal analysis of Birkhoff averages for interval maps with finitely many branches and parabolic fixed points. Using the thermodynamic approach, we strengthen the results of Johansson et al. on the conditional variational principle for the multifractal spectra of Birkhoff averages. To do this, we develop several refined properties of the thermodynamic formalism for non-uniformly expanding interval maps.

math.DS↗

arXiv2025

Bowen's formula and the difference between random iterated function systems and random recursive constructions

In this paper, we show that the difference in independence structures between random iterated function systems and random recursive constructions is reflected in the validity of Bowen's formula. More precisely, we construct random iterated function systems for which Bowen's formula fails, whereas Bowen's formula always holds for random recursive constructions. Our construction demonstrates that the difference in independence structures between the two random models has genuine consequences for dimension theory.

math.DS↗

arXiv2025

Multifractal analysis of the Lyapunov exponent for random graph directed Markov systems

In this paper, we perform the multifractal analysis of the Lyapunov exponent for random conformal graph directed Markov systems introduced by Roy and Urba\'nski (2011). We also generalize Bowen's formula for the limit set of a random conformal graph directed Markov system established by Roy and Urba\'nski. To do this, we develop several refined properties of random finitely primitive countable Markov shifts.

math.DS↗

arXiv2024

Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups

We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which differences in the Hausdorff dimension of the limit set generated by a Markov system cause essentially different results on multifractal analysis. This paper also provides a new characterization of the geodesic flow on the Poinca\'re disc model of two-dimensional hyperbolic space and the limit set of a generalized Schottky group. To prove our main theorem we use thermodynamic formalism on a countable Markov shift, gamma function, and zeta function.

math.DS↗

arXiv2024

Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces

We perform a multifractal analysis of the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with $m \geq 1$ cusps. Our main theorem establishes a conditional variational principle for the Hausdorff dimension spectrum of the multi-cusp winding process. Moreover, we show that the dimension spectrum defined on $\mathbb{R}_{>0}^m$ is real analytic. To prove the main theorem we use a countable Markov shift with a finitely primitive transition matrix and thermodynamic formalism.

math.DS↗