Teturo Kamae

Sequence entropy and the maximal pattern complexity of infinite words

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dots , over a finite alphabet A , we define the maximal pattern complexity 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,\dots\} where the ‘sup’ is taken over all subsequences 0=\tau(0) of integers of length k . We prove that \alpha is eventually periodic if and only if p_\alpha^*(k)\le 2k-1 for some k . Infinite words \alpha , with p_\alpha^*(k)=2k for any k , are called pattern Sturmian words and are studied.

Van der corput’s difference theorem

We obtain a sufficient condition for a subsetH of positive integers to satisfy that the equidistribution (mod 1) of the sequences (un+h− un; n=1, 2, ···) for allh ∈H implies the equidistribution of (un). Our condition is satisfied, for example, for the following sets: (1)H={n − m; n ∈ I, m ∈ I, n>m}, whereI is any infinite subset of integers; (2)H={| ψ (n)|; ψ(n)≠0,n ∈ Z}, where ψ is a nonconstant polynomial with integral coefficients having at least one integral zero (modq) for allq=2, 3, ···; (3)H={p+1;p is a prime} andH={p − 1;p is a prime}.

Maximal pattern complexity for discrete systems

For an infinite word α = α0α1α2 · · · over a finite alphabet, Kamae introduced a new notion of complexity called maximal pattern complexity defined by pα(k) := sup τ ]{αn+τ(0)αn+τ(1) · · ·αn+τ(k−1); n = 0, 1, 2, · · · } where the supremum is taken over all sequences of integers 0 = τ(0) < τ(1) < · · · < τ(k − 1) of length k. Kamae proved that α is aperiodic if and only if pα(k) ≥ 2k for every k = 1, 2, · · · . A word α with pα(k) = 2k for every k ≥ 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 ∗Department of Mathematics, Osaka City University, Osaka, 558-8585 Japan (kamae@sci.osaka-cu.ac.jp) †Department of Mathematics, University of North Texas, Denton, TX 762035116, USA (luca@unt.edu)

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume thatn→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θτ(0), whereθτ(i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.

A characterization of self-affine functions

A characterization of self-affine functions as functions generated by finite automata is given. Also, a kind of uniqueness in representing a self-affine function by a finite automaton is proved. It follows from them that there exist exactly a countably infinite number of self-affine functions modulo constant multiplications.

A simple proof of the ergodic theorem using nonstandard analysis

A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.

Normal numbers and selection rules

AbstractGiven anormal number x=0,x1x2 ··· to base 2 and aselection rule S ⊂{0, 1}*=∪n=0/t8{0, 1}n, we define a subsequencex,=0,

Maximal pattern complexity of words over l letters

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