핵심 개념
Study the density of group languages recognized by morphisms onto finite groups in shift spaces.
초록
The content delves into the density of group languages in shift spaces, focusing on rational languages recognized by morphisms onto finite groups. It explores the ergodicity of skew products and their relation to recognizing groups. The study includes shifts of finite type and minimal shifts, providing formulas for density and conditions for ergodicity. The paper also investigates the link between minimal closed invariant subsets, return words, and bifix codes.
Introduction
Language density traced back to Schützenberger, Berstel, Hansel, and Perrin.
Motivation from symbolic dynamics for studying language density patterns in shift spaces.
Symbolic dynamics
Definitions of factors, recurrence, shift maps, shift spaces, and invariant measures.
Uniquely ergodic shifts and their properties.
Group languages and skew products
Study of density in group languages recognized by morphisms onto finite groups.
Formulas for density and ergodicity in skew products.
Example in the Fibonacci shift illustrating density convergence.
Densities in shifts of finite type
Application of density formulas in shifts of finite type.
Introduction of ϕ-irreducibility and its relation to topological transitivity.
Propositions and theorems establishing conditions for ergodicity and transitivity in skew products.
통계
The formula resembles an earlier result of Hansel and Perrin within the setting of Bernoulli measures ([28], Theorem 3).
The density δµ(L) exists and is given by δµ(L) = |K|/|G|.
인용구
"There are however many cases where the measure ν × µ is not ergodic."
"With an eye on such examples, we establish a more general formula..."