Fractal Dimensions for Poincaré Recurrences

The Rényi entropy function and the large deviation of short return times

“…This quantity arose in several circumstances: -Since it controls the short returns, it plays a crucial role to establish the asymptotic (exponential) distribution of the return times function τ A (x) when the measure of the set A goes to zero [21,3,1,2,26,25,24]. -It has been used to define the recurrence dimension since it served as the gauge set function to construct a suitable Carathéodory measure [5,30,7]. -It has been related to the Algorithmic Information Content in [12].…”

Quantitative Poincaré recurrence in continued fraction dynamical system

“…Here and hereafter, T denotes the Gauss transformation defined by (1). As consequences, every irrational x ∈ (0, 1) can be represented uniquely into its continued fraction expansion where a n (x) = 1 T n−1 x is a positive integer and is called the n-th partial quotient of x.…”

“…The interested readers are referred to Afraimovich et al [1] and the references therein for more information.…”

Local and global approaches to the problem of Poincaré recurrences. Applications in nonlinear dynamics

“…It describes the statistics of recurrence time sequences both in a neighborhood of a given initial state (the local approach or local theory [1,3,19,20,21,22,23,24,25,26,27]) and in the considered set of phase trajectories of the system (the global approach or global theory [28,29,30,31,32,33,34,35]). It has been proven that the mean recurrence time in a neighborhood of a given state is interrelated with the probability of the phase trajectory visiting this neighborhood (Kac's lemma [19,20]).…”

“…This approach deals with the mean recurrence time which is found over all elements of the full covering of the considered set of phase trajectories [29,32]. In this case, the mean recurrence time depends on a sequence of initial points specified in each covering element of the set, and it is a function of the whole set.…”