arXiv subjects

Maxim Kirsebom

Publications and source records attributed to Maxim Kirsebom.

8 recordsLinked to original sources

arXiv2026

Visiting time statistics

Many mixing dynamical systems $(X,T,\mu)$ are known to satisfy the hitting time statistics result \[ \lim_{r \to 0} \mu \{\, x : \tau_{B(y,r)} (x) > t/\mu(B(y,r))\,\} = e^{-t}, \] for $\mu$-almost every $y$, where $\tau_{B(y,r)} (x)$ is the first hitting time of $x$ to the ball $B(y,r)$. Taking a different point of view, we fix $x$ and consider $\tau_{B(y,r)} (x)$ as a function of $y$. We call this the visiting time of $y$ from $x$, i.e. the time it takes for $y$ to get a visit from $x$ within a neighbourhood of radius $r$. We prove that \[ \lim_{r \to 0} \mu \{\, y : \tau_{B(y,r)} (x) > t/\mu(B(y,r)) \,\} = e^{-t}, \] for $\mu$-almost every $x$. As a byproduct we obtain a new method of proof for hitting time statistics.

math.DS↗

arXiv2025

Almost sure orbits closeness

We consider the minimal distance between orbits of measure preserving dynamical systems. In the spirit of dynamical shrinking target problems we identify distance rates for which almost sure asymptotic closeness properties can be ensured. More precisely, we consider the set $E_n$ of pairs of points whose orbits up to time $n$ have minimal distance to each other less than the threshold $r_n$. We obtain bounds on the sequence $(r_n)_n$ to guarantee that $\limsup_{n}E_n$ and $\liminf_{n} E_n$ are sets of measure 0 or 1. Results for the measure 0 case are obtained in broad generality while the measure one case requires assumptions of exponential mixing for at least one of the systems. We also consider the analogous question of the minimal distance of points within a single orbit of one dimensional exponentially mixing dynamical systems.

math.DS↗

arXiv2022

On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$

We study an extreme value distribution for the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for $F(r)$ for $r \in [-\frac{1}{2} \log 2, \infty)$, and establish asymptotic behavior of $F(r)$ as $r \to -\infty$.

math.DS↗

arXiv2022

Extreme Value Theory for Hurwitz Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we get several results concerning extremes of nearest integer continued fractions as well.

math.NT↗

arXiv2021

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes

Suppose $(f,\mathcal{X},μ)$ is a measure preserving dynamical system and $ϕ\colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=ϕ\circ f^{i-1}$ is a time series of observations on the system. Suppose that $(u_n)$ is a non-decreasing sequence of real numbers, such that $μ(X_1>u_n)\to 0$. For certain dynamical systems, we obtain a zero--one measure dichotomy for $μ(M_n\leq u_n\,\textrm{i.o.})$ depending on the sequence $u_n$. Specific examples are piecewise expanding interval maps including the Gauss map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences $u_n$. Our results on the permitted sequences $u_n$ are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

math.DS↗

arXiv2021

On shrinking targets and self-returning points

We consider the set $\mathcal{R}_\mathrm{io}$ of points returning infinitely many times to a sequence of shrinking targets around themselves. Under additional assumptions we improve Boshernitzan's pioneering result on the speed of recurrence. In the case of the doubling map as well as some linear maps on the $d$ dimensional torus, we even obtain a dichotomy condition for $\mathcal{R}_\mathrm{io}$ to have measure zero or one. Moreover, we study the set of points eventually always returning and prove an analogue of Boshernitzan's result in similar generality.

math.DS↗

arXiv2019

Continued fractions, the Chen-Stein method and extreme value theory

In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

math.PR↗

arXiv2019

Shrinking targets and eventually always hitting points for interval maps

We study shrinking target problems and the set $\mathcal{E}_{\text{ah}}$ of eventually always hitting points. These are the points whose first $n$ iterates will never have empty intersection with the $n$-th target for sufficiently large $n$. We derive necessary and sufficient conditions on the shrinking rate of the targets for $\mathcal{E}_{\text{ah}}$ to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville-Pomeau map. We also obtain results for the Gauss map and correspondingly for the maximal digits in continued fractions expansions. In the case of the doubling map we also compute the packing dimension of $\mathcal{E}_{\text{ah}}$ complementing already known results on the Hausdorff dimension of $\mathcal{E}_{\text{ah}}$.

math.DS↗