arXiv subjects

Shunsuke Usuki

Publications and source records attributed to Shunsuke Usuki.

5 recordsLinked to original sources

arXiv2025

On the $L^q$ dimension of stationary measures for Möbius iterated function systems

We study the $L^q$ dimension $D(ν,q)\ (q>1)$ of stationary measures $ν$ for Möbius iterated function systems on $\mathbb{R}$ satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem 6.6]{Shm19}. As the result, we show that there is the dichotomy: the $L^q$ spectrum $τ(ν,q)=(q-1)D(ν,q)$ is equal to the desired value $\min\{\widetildeτ(ν,q),q-1\}$ for any $q>1$, where $\widetildeτ(ν,q)$ is the zero of the canonical pressure function, or there exist $q_0>1$ and $0<α<1$ such that $τ(ν,q)=\min\{\widetildeτ(ν,q),q-1\}$ for $1<q<q_0$ and $τ(ν,q)=αq$ for $q\geq q_0$. In addition, we give examples of Möbius iterated function systems which show the latter case by giving an affirmative answer to Solomyak's question \cite[Question 2]{Sol24}.

math.DS↗

arXiv2024

An improvement of the lower bound of the number of integers in Littlewood's conjecture

In this paper, we improve the results in the author's previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood's conjecture. We show that, for any $0<γ<1$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ(\log N)^2/(\log\log N)^2$ up to a universal constant.

math.NT↗

arXiv2022

On a lower bound of the number of integers in Littlewood's conjecture

We show that, for any $0<γ<1/2$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ\varepsilon\log N$ up to a uniform constant. This can be seen as a quantitative result on the fact that the exceptional set to Littlewood's conjecture has Hausdorff dimension zero, obtained by M. Einsiedler, A. Katok and E. Lindenstrauss in 2000's. For the proof, we study the behavior of the empirical measures with respect to the diagonal action on $\rm{SL}(3,\mathbb{R})/\rm{SL}(3,\mathbb{Z})$ and show that we can obtain a quantitative result on Littlewood's conjecture for $(α,β)$ if the corresponding empirical measures are well-behaved. We also estimate Hausdorff dimension of the exceptional set to be small.

math.NT↗

arXiv2022

$\times a$ and $\times b$ empirical measures, the irregular set and entropy

For integers $a$ and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by $a$ and $b$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. The action on $\mathbb{T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if $a$ and $b$ are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, whether there exists a nontrivial $\times a,\times b$ invariant and ergodic measure is not known. In this paper, we study the empirical measures of $x\in\mathbb{T}$ with respect to the $\times a,\times b$ action and show that the set of $x$ such that the empirical measures of $x$ do not converge to any measure has Hausdorff dimension $1$ and the set of $x$ such that the empirical measures can approach a nontrivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of $x$ in the complement of a set of Hausdorff dimension zero.

math.DS↗

arXiv2022

Stability of uniqueness and coexistence of equilibrium states of the Ising model under long range perturbations

In this paper, we study perturbations of the $d$-dimensional Ising model for $d\geq 2$, including long range ones to which the Pirogov-Sinai theory is not applicable. We show that the uniqueness of the equilibrium state of the Ising model at high temperature and the coexistence of equilibrium states at low temperature are preserved by spin-flip symmetric perturbations.

math-ph↗