Nonlinear Maps and their Applications by Clara Grácio Daniele Fournier-Prunaret Tetsushi Ueta & Yoshifumi Nishio

Author:Clara Grácio, Daniele Fournier-Prunaret, Tetsushi Ueta & Yoshifumi Nishio
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

8.3.2 The ∗-Product

A pair (X, Y ) ∈ Σ×Σ is admissible if X 0 = L, Y 0 = R, and for Z ∈ X, Y and all k > 0, 1. Z k  = L ⇒ s k (Z) ≤ X;

2. Z k  = R ⇒ Y ≤ s k (Z).

Each of these inequalities is strict if, respectively, X or Y is finite. It can also be shown that a pair of periodic sequences is admissible if and only if the associated pair of finite sequences is admissible.

These properties justify the usual identification between periodic maximal or minimal sequences with their finite counterparts (X 0 … X n−10). Both can be identified by the finite aperiodic word X 0 … X n−1, which we will do whenever appropriate.

The ∗-product between a pair of finite sequences (X, Y ) and another finite sequence U is defined as

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