arXiv subjects

Alejandro Rodriguez Sponheimer

Publications and source records attributed to Alejandro Rodriguez Sponheimer.

4 recordsLinked to original sources

arXiv2025

Limit Laws for Poincar\'e Recurrence and the Shrinking Target Problem

We establish distributional laws for Poincar\'e recurrence in measure-preserving systems $(X,T,\mu)$ satisfying an exponential multiple decorrelation condition and a short returns condition. When the measure is absolutely continuous, the sum $\sum_{k=1}^{n} \mathbf{1}_{B(x,r_k)}(T^{k}x) - \mu(B(x,r_k))$ does not in general obey a CLT; instead, it converges to a non-standard distribution that is an average of Gaussian laws weighted by the density of $\mu$. By considering a version of the sum where we appropriately rescale the radii of the balls, we recover the CLT. A key assumption in our recurrence theorems is that the corresponding hitting sums satisfy the CLT. We verify this assumption for Axiom A systems by establishing the stronger ASIP for the shrinking target problem, extending Haydn, Nicol, T\"or\"ok and Vaienti [Trans. Amer. Math. Soc. 2017] and related results. Systems for which our results apply include piecewise expanding systems on the interval, and Axiom A systems. The results highlight the difference between recurrence and hitting behaviour.

math.DS↗

arXiv2025

Strong Borel--Cantelli Lemmas for Recurrence

Let $(X,T,\mu,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there exist open balls $B_k(x)$ around $x$ such that $\mu(B_k(x)) = M_k$. Under a short return time assumption, we prove a strong Borel--Cantelli lemma, including an error term, for recurrence, i.e., for $\mu$-a.e. $x \in X$, \[ \sum_{k=1}^{n} \mathbf{1}_{B_k(x)} (T^k x) = \Phi(n) + O \bigl( \Phi(n)^{1/2} (\log \Phi(n))^{3/2 + \varepsilon} \bigr), \] where $\Phi(n) = \sum_{k=1}^{n} \mu(B_k(x))$. Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of $\mathbf{T}^2$.

math.DS↗

arXiv2024

On uniform recurrence for hyperbolic automorphisms of the $2$-dimensional torus

We are interested in studying sets of the form \[ \mathcal{U}(α) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |λ|^{-αN} \right\} \] where $(X,T,d)$ is our metric dynamical system and $|λ|>1$. Although a lot of results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider $X=\mathbb{T}^2$, $T(x) = Ax \pmod{1}$, where $A$ is a hyperbolic, area preserving, $2\times 2$ matrix with integer entries and $λ$ is the eigenvalue of $A$ of modulus larger than $1$ and we explicitly calculate the Hausdorff dimension of this set.

math.DS↗

arXiv2023

A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms

Let $(X,\mu,T,d)$ be a metric measure-preserving dynamical system such that $3$-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel--Cantelli result for recurrence, i.e., for $\mu$-a.e. $x\in X$ \[ \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \] where $\mu(B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.

math.DS↗