Shortest Distance Between Multiple Orbits and Generalized Fractal Dimensions

Closest distance between iterates of typical points

“…, k − 1, for some i 1 , i 2 ≤ n − k} converges to a Rényi entropy for µ ⊗ 2 − almost every (x, y) in Σ 2 A . Similar almost sure convergences are proved for k−point-orbits in [4] for all k ≥ 2.…”

Rényi entropy and pattern matching for run-length encoded sequences

“…In Barros et al (2019), it was proved that for α-mixing process with exponential decay M n ∼ 2 H2(µ) log n almost surely, where H 2 (µ) is the Rényi entropy (see Definition 2.3) of the stationary measure µ. Similar results have been proved for more than two sequences Barros and Rousseau (2021) and for random sequences in random environment Rousseau (2021+).…”

Matching of observations of dynamical systems, with applications to sequence matching

“…When it comes to written text, a complex dependence structure can arise from specic sequences of letters, such as words, and higher-order syntactic and narrative structures. Recently, Barros, Liao and Rousseau adopted a dynamical system point of view to give the asymptotic behavior of the length of the longest sub-sequence common to dierent α−mixing sequences [7,8]. This problem is dierent from the present one, because the sub-sequences may be present at dierent locations of the dierent strings of symbols, but we will also follow a dynamical system approach to derive our results.…”

“…Closely related problems have gained interest in recent years. The case of real, unobserved trajectories was considered in [19] and [11], using EVT techniques, while asymptotic results for the shortest distance between two orbits were obtained in [7] and then generalized to multiple orbits [8] and finally to observed orbits [14].…”