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What is the Golden Mean Shift?

Published in Symbolic Dynamics 2 mins read

The golden mean shift is a specific mathematical construct within the field of symbolic dynamics, precisely defined as the collection of all binary sequences that do not contain two consecutive ones.

Defining the Golden Mean Shift

More formally, it is identified as a shift space comprised solely of sequences made up of the symbols '0' and '1'. What distinguishes it is a strict rule: the pattern '11' is explicitly forbidden from appearing anywhere within any sequence belonging to this set. This means it is considered a subshift where the set of forbidden words, denoted as F, contains only the word '11' (i.e., F = {11}).

This concept is foundational in understanding various dynamical systems by representing their behaviors as sequences of symbols, where specific patterns are either permitted or restricted.

Characteristics and Rules

  • Binary Sequences: All sequences consist only of 0s and 1s.
  • Forbidden Pattern: The defining characteristic is the absolute absence of '11'. Any sequence containing '11' is excluded from the golden mean shift.
  • Infinite Length: Typically, these sequences are considered infinitely long in both directions (bi-infinite).

Examples of Golden Mean Shift Sequences

To illustrate, consider the following examples of binary sequences and whether they belong to the golden mean shift:

Sequence Belongs to Golden Mean Shift? Reason
...0101010... Yes Contains no '11'
...1010101... Yes Contains no '11'
00000 Yes Contains no '11'
10010 Yes Contains no '11'
11010 No Contains the forbidden pattern '11'
01100 No Contains the forbidden pattern '11'
10110 No Contains the forbidden pattern '11'

This shift space is significant because its properties, such as its topological entropy, are directly related to the golden ratio, which gives it its name. It provides a simple yet rich example for studying complex system behaviors through symbolic representation within symbolic dynamics.