arXiv subjects

T. Persson

Publications and source records attributed to T. Persson.

3 recordsLinked to original sources

arXiv2023

On the distribution of sequences of the form $(q_ny)$

We study the distribution of sequences of the form $(q_ny)_{n=1}^\infty$, where $(q_n)_{n=1}^\infty$ is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points $\gamma \in [0,1)$ which are well approximated by points in the sequence $(q_ny)_{n=1}^\infty$. The bounds on Hausdorff dimension are valid for almost every $y$ in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is itself Lebesgue measure, our measure bounds are also sharp for a very large class of sequences. We also give an application to inhomogeneous Littlewood type problems.

math.NT↗

arXiv2019

On the fractional susceptibility function of piecewise expanding maps

We associate to a perturbation $(f_t)$ of a (stably mixing) piecewise expanding unimodal map $f_0$ a two-variable fractional susceptibility function $\Psi_\phi(\eta, z)$, depending also on a bounded observable $\phi$. For fixed $\eta \in (0,1)$, we show that the function $\Psi_\phi(\eta, z)$ is holomorphic in a disc $D_\eta\subset \mathbb{C}$ centered at zero of radius $>1$, and that $\Psi_\phi(\eta, 1)$ is the Marchaud fractional derivative of order $\eta$ of the function $t\mapsto \mathcal{R}_\phi(t):=\int \phi(x)\, d\mu_t$, at $t=0$, where $\mu_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we show that $\Psi_\phi(\eta, z)$ admits a holomorphic extension to the domain $\{ (\eta, z) \in {\mathbb{C}}^2\mid 0<\Re \eta <1, \, z \in D_\eta \}$. Finally, if the perturbation $(f_t)$ is horizontal, we prove that $\lim_{\eta \to 1}\Psi_\phi(\eta, 1)=\partial_t \mathcal{R}_\phi(t)|_{t=0}$.

math.DS↗

arXiv2011

A note on Borel--Cantelli lemmas for non-uniformly hyperbolic dynamical systems

Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\mathcal{B}, μ)$ such that $\sum_{n=1}^{\infty} μ(B_{i}) = \infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $μ(\{x \in X : x \in B_{i} \text{infinitely often (i.o.)}) = 1$. Suppose $(T,X,μ)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\in B_i$ for $μ$ a.e.\ $x\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\in B_i$ infinitely often for $μ$ a.e.\ $x$ we call the sequence $B_i$ a Borel--Cantelli sequence. If the sets $B_i:= B(p,r_i)$ are nested balls about a point $p$ then the question of whether $T^i x\in B_i$ infinitely often for $μ$ a.e.\ $x$ is often called the shrinking target problem. We show, under certain assumptions on the measure $μ$, that for balls $B_i$ if $μ(B_i)\ge i^{-γ}$, $0<γ<1$, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observations implies that the sequence is Borel-Cantelli. If $μ(B_i)\ge \frac{C\log i}{i}$ then exponential decay of correlations implies that the sequence is Borel-Cantelli. If it is only assumed that $μ(B_i) \ge \frac{1}{i}$ then we give conditions in terms of return time statistics which imply that for $μ$ a.e.\ $p$ sequences of nested balls $B(p,1/i)$ are Borel-Cantelli. Corollaries of our results are that for planar dispersing billiards and Lozi maps $μ$ a.e.\ $p$ sequences of nested balls $B(p,1/i)$ are Borel-Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

math.DS↗