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Romain Aimino

Publications and source records attributed to Romain Aimino.

8 recordsLinked to original sources
arXiv2026

Distributional results for the shortest distance between trajectories of different dynamics

We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit densities with isolated zeros and singularities. We present diverse examples of systems satisfying these assumptions and compute the different parameters modulating the limit distribution.

math.DS↗
arXiv2026

Multivariate extreme values for dynamical systems

We establish a theory for multivariate extreme value analysis of dynamical systems. Namely, we provide conditions adapted to the dynamical setting which enable the study of dependence between extreme values of the components of $\R^d$-valued observables evaluated along the orbits of the systems. We study this cross-sectional dependence, which results from the combination of a spatial and a temporal dependence structures. We give several illustrative applications, where concrete systems and dependence sources are introduced and analysed.

math.DS↗
arXiv2024

Stable laws for random dynamical systems

In this paper we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an iid fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $ν$. We consider a class of non square-integrable observables $ϕ$, mostly of form $ϕ(x)=d(x,x_0)^{-\frac{1}α}$ where $x_0$ is non-periodic point satisfying some other genericity conditions, and more generally regularly varying observables with index $α\in (0,2)$. The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps, and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $α$ we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law and functional stable limit laws, in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $ν$. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

math.DS↗
arXiv2020

Deterministic walks in random environment

Motivated by the random Lorentz gas, we study deterministic walks in random environment and show that (in simple, yet relevant, cases) they can be reduced to a class of random walks in random environment where the jump probability depends (weakly) on the past. In addition, we prove few basic results (hopefully the germ of a general theory) on the latter, purely probabilistic, model.

math.PR↗
arXiv2016

Recurrence statistics for the space of Interval Exchange maps and the Teichmüller flow on the space of translation surfaces

In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Hölder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmüller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in Hölder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Hölder function space is particularly useful for investigating return time statistics. In particular we establish the shrinking target property for nested balls in the setting of Teichmüller flow. Our point of view, approach and terminology derives from the work of M. Pollicott augmented by that of M. Viana.

math.DS↗
arXiv2015

Concentration inequalities for sequential dynamical systems of the unit interval

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BV norm, and we investigate several of its consequences. In particular, this covers compositions of $β$-transformations, with all $β$ lying in a neighborhood of a fixed $β_{\star} > 1$ and systems satisfying a covering type assumption.

math.DS↗
arXiv2014

Annealed and quenched limit theorems for random expanding dynamical systems

In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erdös-Rényi laws and concentration inequalities.

math.DS↗