Stable laws for random dynamical systems, (.pdf)
R. Aimino, M. Nicol, A. Török, (preprint), (2022)
Abstract: 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 \(\nu\). We consider a class of non square-integrable observables \(\phi\), mostly of form \(\phi(x)=d(x,x_0)^{-\frac{1}{\alpha}}\) where \(x_0\) is non-periodic point satisfying some other genericity conditions, and more generally regularly varying observables with index \(\alpha \in (0,2)\). 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 \(\alpha\) 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 \(\nu\). 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.
Statistical properties of the Rauzy-Veech-Zorich map, (.pdf)
R. Aimino, M. Pollicott, Thermodynamic Formalism, Lectures Notes in Mathematics, CIRM Jean-Morlet subseries, 2290, 317-349, Pollicott M., Vaienti S. (eds.), Springer, Cham., (2021), (link)
Abstract: In this note we survey some very basic statistical properties of the Rauzy-Veech map and the Zorich acceleration. Our aim is to give a particularly thermodynamic perspective of well known results.
Deterministic walks in random environment, (.pdf)
R. Aimino, C. Liverani, Annals of Probability, 48 (5), 2212-2257, (2020), (link)
Abstract: 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.
Large deviations for dynamical systems with stretched exponential decay of correlations, (.pdf)
R. Aimino, J. M. Freitas, Portugaliae Mathematica, 76 (2), 143-152, (2019), (link)
Recurrence statistics for the space of Interval Exchange maps and the Teichmüller flow on the space of translation surfaces, (.pdf)
R. Aimino, M. Nicol, M. Todd, Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 53 (3), 1371-1401, (2017), (link)
Abstract: 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.
On the quenched central limit theorem for random dynamical systems, (.pdf)
M. Abdelkader, R. Aimino, Journal of Physics A: Mathematical and Theoretical, 49 (24), 13pp, (2016), (link)
Abstract: We provide a necessary and sufficient condition under which the quenched central limit theorem without random centering holds for one-dimensional random systems that are uniformly expanding. This condition holds in particular when all the maps preserve a common measure. We also give a counter example which shows that this condition is not necessarily satisfied when the maps do not preserve a common measure.
Concentration inequalities for sequential dynamical systems of the unit interval, (.pdf)
R. Aimino, J. Rousseau, Ergodic Theory and Dynamical Systems, 36 (8), 2384-2407, (2016), (link)
Abstract: 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 \(\beta\)-transformations, with all \(\beta\) lying in a neighborhood of a fixed \(\beta_\star > 1\) and systems satisfying a covering type assumption.
Polynomial loss of memory for maps of the interval with a neutral fixed point, (.pdf)
R. Aimino, H. Hu, M. Nicol, A. Török, S. Vaienti, Discrete and Continuous Dynamical Systems -A, 35 (3), 793-806, (2015), (link)
Abstract: We give an example of a sequential dynamical system consisting of intermittent-type maps which exhibits loss of memory with a polynomial rate of decay. A uniform bound holds for the upper rate of memory loss. The maps may be chosen in any sequence, and the bound holds for all compositions.
Annealed and quenched limit theorems for random expanding dynamical systems, (.pdf)
R. Aimino, M. Nicol, S. Vaienti, Probability Theory and Related Fields, 162 (1), 233-274, (2015), (link)
Abstract: 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.
A note on the large deviations for piecewise expanding multidimensional maps, (.pdf)
R. Aimino, S. Vaienti, Nonlinear Dynamics, New Directions: Theoretical Aspects, Nonlinear Systems and Complexity, 11, González-Aguilar, H. and Ugalde, E (eds.), New York, Springer. 2015, (link)
Abstract: We provide the large deviation principle for some systems admitting a spectral gap, by using the functional approach of Hennion and Hervé, slightly modified. Our main application concerns multidimensional expanding maps introduced by Saussol.
Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes, (.pdf)
R. Aimino, Thèse de doctorat, Université de Toulon, (2014)
Abstract: Dans cette thèse, nous nous intéressons aux propriétés statistiques des systèmes dynamiques aléatoires et non-autonomes. Dans le premier chapitre, consacré aux systèmes aléatoires, nous établissons un cadre fonctionnel abstrait, couvrant une large classe de systèmes dilatants en dimension 1 et supérieure, permettant de démontrer de nombreux théorèmes limites annealed. Nous donnons aussi une condition nécessaire et suffisante pour que la version quenched du théorème de la limite centrale soit valide en dimension 1. Dans le chapitre deux, après avoir introduit la notion de système non-autonome, nous étudions un système composé d’applications en dimension 1 ayant un point fixe neutre commun, et nous montrons que celui-ci admet une vitesse de perte de mémoire polynomiale. Le chapitre trois est consacré aux inégalités de concentration. Nous établissons de telles inégalités pour des systèmes dynamiques aléatoires et non-autonomes, et nous étudions diverses applications. Dans le chapitre quatre, nous nous intéressons aux lemmes dynamiques de Borel-Cantelli pour l’induction de Rauzy-Veech-Zorich, et présentons quelques résultats liés aux statistiques de récurrence pour cette application.