Maximal pattern complexity for discrete systems

Abstract

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dotsc over a finite alphabet, the authors introduced a new notion of complexity called maximal pattern complexity defined by p_\alpha^*(k):=\sup_\tau\sharp\{\alpha_{n+\tau(0)}\alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)};n=0,1,2,\dotsc\} where the supremum is taken over all sequences of integers 0=\tau(0)<\tau(1)<\dotsc<\tau(k-1) of length k. The authors proved that \alpha is aperiodic if and only if p_\alpha^*(k)\ge 2k for every k=1,2,\dotsc. A word \alpha with p_\alpha^*(k)=2k for every k\geq 1 is called pattern Sturmian. In this paper, we give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations. We also investigate the maximal pattern complexity of various discrete dynamical systems including irrational rotations on the circle, and self-similar systems generated by substitutions. We show that, for each irrational rotation on the circle, there exists a twofold partition of the circle, with respect to which the system generated has full maximal pattern complexity with probability one. Using the arithmetic properties of the underlying numeration system associated to a substitution dynamical system, we prove that the maximal pattern complexity of the fixed point of the Rauzy substitution 1\mapsto 12, 2\mapsto 13, 3\mapsto 1 has exponential growth. It is well known that the system generated by the Rauzy substitution is isomorphic in measure to an irrational rotation on the 2-torus.