#### Date of Award

1-1-2016

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Ronald Pavlov, Ph.D.

#### Keywords

Extender sets, Follower sets, Symbolic dynamics

#### Abstract

Given a word *w* in the language of a one-dimensional shift space *X*, the follower set of *w*, denoted *F _{X}(w)*, is the set of all right-infinite sequences which follow

*w*in some point of

*X*. Extender sets are a generalization of follower sets and are defined similarly. To a given shift space

*X*, then, we may associate a follower set sequence {|

*F*

_{X}*(n)*|} which records the number of distinct follower sets in

*X*corresponding to words of length

*n*. Similarly, we may define an extender set sequence {|

*E*|}. The complexity sequence {Phi

_{X}(n)*} of a shift space*

_{X}(n)*X*records the number of n-letter words in the language of

*X*for each

*n*. This thesis explores the relationship between the class of achievable follower and extender set sequences of one-dimensional shift spaces and the class of their complexity sequences.

Some surprising similarities suggest a connection may exist, for instance, both the complexity sequence and the extender set sequence are bounded if and only if there exists some *n* such that the value of the *n ^{th}* term of the sequence is at most

*n*. This thesis, however, also demonstrates important differences among complexity sequences and follower and extender set sequences of one-dimensional shifts. In particular, we show that unlike complexity sequences, follower and extender set sequences need not be monotone increasing. Finally, we use the classical

*beta*-shifts to demonstrate that, while many follower set sequences may not be realized as complexity sequences, up to possible increase by 1, any complexity sequence may be realized as a follower set sequence of some shift space.

#### Recommended Citation

French, Thomas Kelly, "Follower and Extender Sets in Symbolic Dynamics" (2016). *Electronic Theses and Dissertations*. 1152.

https://digitalcommons.du.edu/etd/1152

#### Provenance

Received from ProQuest

#### Rights holder

Thomas Kelly French

#### File size

91 p.

#### Copyright date

2016

#### File format

application/pdf

#### Language

en

#### Discipline

Mathematics

## Comments

Copyright is held by the author.