Convergence and Factor Complexity for the Arnoux-Rauzy-Poincaré Algorithm

Abstract

Abstract. We introduce a multidimensional continued fraction algorithm based on Arnoux-Rauzy and Poincaré algorithms, and we study its associated S-adic system. An S-adic system is made of infinite words generated by the composition of infinite sequences of substitutions with values in a given finite set of substitutions, together with some restrictions concerning the allowed sequences of substitutions, expressed in terms of a regular language. We prove that these words have a factor complexity p(n) with lim sup p(n)/n < 3, which provides a proof for the convergence of the associated algorithm by unique ergodicity.

Factor complexity of S-adic words generated by the Arnoux–Rauzy–Poincaré algorithm

“…The convergence of the algorithm together with unique ergodicity is lastly considered in Section 6. This article is an extended version of [BL13]. The present paper provides the upper bound p(n) ≤ 5 2 n + 1, whereas the upper bound in [BL13] was p(n) ≤ 3n + 1.…”

“…This article is an extended version of [BL13]. The present paper provides the upper bound p(n) ≤ 5 2 n + 1, whereas the upper bound in [BL13] was p(n) ≤ 3n + 1.…”

Dynamical properties of simplicial systems and continued fraction algorithms

“…Ergodicity for Brun and Selmer algorithms in all dimension is due to Schweiger [Sch00], for Arnoux-Rauzy-Poincaré it has been proved in [BL13]. The result on Hausdorff dimension has been proved in dimension 2 in [AHS16].…”

“…For more than 30 years, a large community of mathematicians have been working on proving dynamical properties of MCF, such as convergence [Fis72], [Nog95], [BL13], as well as further dynamical properties like ergodicity [Sch90], [MNS09], [BFK15], construction of invariant measures [AL18], [AS17] and estimates on the speed of convergence through Lyapunov exponents [Lag93], [BAG01], [FS19].…”