Potential Well in Poincaré Recurrence

Abstract

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

Return-time -spectrum for equilibrium states with potentials of summable variation

“…product measure) with m being the Bernoulli distribution (that is, A = {0, 1}) with parameter p = 1/3. This corresponds to a potential ϕ which is locally constant on the cylinders [0] and [1], and therefore it obviously fulfils the conditions of the theorem. See Subsection 3.4).…”

“…As already mentioned, equilibrium states with potential of summable variation are -mixing (see Proposition 4.1). Since, moreover, they have complete grammar, they, therefore, satisfy the conditions of Theorem 2 of [1]. This result states that the potential well is bounded away from , that is, in which .…”

“…In [1], this is Theorem 1, statement 2, combined with Remark 2. A consequence of ψ-mixing is that there exist c 1 , c 2 > 0 such that μ([x n−1 0 ]) ≤ c 1 e −c 2 n for all x and n. This also follows from (5) since ϕ < 0.…”

“…Take, for instance, A = {0, 1} and let μ = m N , where m is a Bernoulli measure on A with parameter p 1 = 1 2 . This corresponds to a potential ϕ which is locally constant on the cylinders [0] and [1]. We can identify it with a function from A to R such that ϕ(1) = log p 1 .…”

“…For , an important ingredient of the proof is an approximation of the distribution of by an exponential law, with a precise error term, recently proved in [1]. Using this result, the computation of is straightforward.…”