Fibonacci-like sequences and shift spaces in symbolic dynamics

TL;DR

This paper explores Fibonacci-like sequences in symbolic dynamics, providing methods to count blocks, compute entropy, and construct shift spaces with desired entropy levels.

Contribution

It introduces a novel approach linking Fibonacci-like recurrences to the calculation of topological entropy in shift spaces.

Findings

Derived recurrence relations for block counting

Established formulas for entropy calculation

Proposed a scheme to design shift spaces with specific entropy

Abstract

We afford the problem of counting the blocks of a given length made with symbols drawn from an alphabet and relate this number to Fibonacci-like recurrent relations. The recurrence polynomia allows to calculate the limit ratio of two adjacent terms and this information is used to determine the topological entropy of a discrete numbers of associate shift spaces. We then describe a scheme to build a shift space with a pre-selected entropy.