Spatio-temporal Poisson processes for visits to small sets
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SPATIO-TEMPORAL POISSON PROCESSES FOR VISITS TO SMALL SETS
BY
Franc ¸ oise P` ene∗ and Benoˆıt Saussol Laboratoire de Math´ematiques de Bretagne Atlantique, CNRS UMR 6205 Univ Brest, Universit´e de Brest, Brest, France e-mail: [email protected], [email protected]
ABSTRACT
For many measure preserving dynamical systems (Ω, T, μ) the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times n of visits to a set A, but also the position T n (x) in A of the orbit, in the limit where μ(A) → 0. We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.
1. Introduction The study of Poincar´e recurrence in dynamical systems such as occurrence of rare events, distribution of return time, hitting time and Poisson law has grown to an active field of research, in deep relation with extreme values of stochastic processes (see [12] and references therein). ∗ Fran¸coise P` ene is supported by the IUF.
Received March 20, 2018 and in revised form July 23, 2019
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` F. PENE AND B. SAUSSOL
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Isr. J. Math.
Let (Ω, F , μ, T ) be a probability preserving dynamical system. For every A ∈ F , we set τA as the first hitting time to A, i.e., τA (x) := inf{n ≥ 1 : T n x ∈ A}. In many systems the behavior of the successive visits of a typical orbit (T n x)n to the sets Aε , with μ(Aε ) → 0+, is often asymptotic, when suitably normalized, to a Poisson process. Such results were first obtained by Doeblin [6] for the Gauss map; Pitskel [16] considered the case of Markov chains. The most recent developments concern non-uniformly hyperbolic dynamical systems, for example [19, 3, 8, 9, 10] just to mention a few of them. An important issue of our work is that we take into account not only the times of successive visits to the set, but also the position of the successive visits in Aε within each return. This study was first motivated by a question posed to us by D. Sz´asz and I. P. T´ oth for diamond billiards, that we address in Section 6. Beyond its own interest, Poisson limit theorems for such spatiotemporal processes have been recently used to prove convergence to L´evy stable processes in dynamical systems; see [20] and subsequent works such as [13]. We note that, at the same time and independently of the present work, analogous processes have been investigated in [7]. We thus consider these events in time and space, {(n, T nx) : n ≥ 1, T n x ∈ Aε } ⊂ [0, +∞) × Ω, that we will normalize both in time and space, as follows:
T 9x 6
Tx T 8x
Aε
T 10x
Tx x
7
T 4x
Tx
3
Tx
5
2
Tx
Tx
Ω
Vol. TBD, 2020
SPATIO-TEMPORAL POISSON PROCESSES
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The successive visit times have order 1/μ(Aε ), whi
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