Symbol ratio minimax sequences in the lexicographic order

Abstract

Consider the space of sequences of k letters ordered lexicographically. We study the set M({\alpha}) of all maximal sequences for which the asymptotic proportions {\alpha} of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of M({\alpha}) is called the {\alpha}-infimax sequence, or the {\alpha}-minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the {\alpha}-infimax for every {\alpha} in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.Comment: 26 pages. Corrected proof of Theorem 23(b) (previously Theorem 21(b)). Modified after discovering connections with work of Bruin and Troubetzko

Geometric representation of the infimax S-adic family

“…Its existence can be obtained from the Multiplicative Ergodic Theorem for almost every sequence with respect to a shift invariant measure µ on Σ + ∞ as long as B n is µ-log integrable. (b) As mentioned in the introduction, the solution to the infimax problem for all asymptotic symbol frequency vectors requires the inclusion of the n = 0 substitution in the family (1.1) ( [11]). However when it is included the family is no longer eventually positive and the analysis requires many special cases and results without much significant additional payoff.…”

“…In [11] an S-adic family was found to generate the solutions to the following problem. A one-sided sequence is called maximal if it is larger in the lexicographic order than all its shifts.…”

“…Returning to the infimax problem, for a given asymptotic digit frequency vector ρ, it was shown in [11] that a three dimensional continued fraction algorithm using ρ as an input generates a list n ∈ Σ + ∞ . The infimax for ρ is then the corresponding α − → as defined in (1.2).…”

“…Another special feature is the central role of the lexicographic order on the symbol space and its relation under the representation map to the usual linear order on the real line. Note that each substitution Λ n preserves the lexicographic order (Lemma 2 in [11]) and that the family arose as the solution to the infimax question which depends fundamentally on the lexicographic order. In the final analysis it is the relation of the orders on sequences and the reals which yields the fundamental fact of the disjointness of the subpieces of the Rauzy fractal.…”

New rotation sets in a family of torus homeomorphisms

“…In this section we will construct a continuously varying family {Φ t } of self-homeomorphisms of the torus whose rotation sets ρ(t) satisfy ρ(t) = ρ 8 (t) for all t ∈ [0, 1]. To do this, we use Theorem 3.1 of [5] to "unwrap" the family f t . This theorem is a generalization of a result of Barge and Martin [4] to parameterized families.…”