Date of Award

Spring 6-12-2026

Document Type

Dissertation

Degree Name

Ph.D. in Mathematics

Organizational Unit

College of Natural Science and Mathematics, Mathematics

First Advisor

Ronnie Pavlov

Second Advisor

Nicholas Ormes

Third Advisor

Anh N. Le

Fourth Advisor

Mei Yin

Copyright Statement / License for Reuse

All Rights Reserved
All Rights Reserved.

Keywords

maximal pattern Complexity, Symbolic dynamics

Abstract

For a finite alphabetΒ π’œ and a sequence π‘₯ ∈ π’œβ„•βŠ¬ , Kamae and Zamboni combined the ideas of block complexity and topological sequence entropy to define the maximal pattern complexity, π‘βˆ—π‘› (π‘₯). They defined an aperiodic sequence π‘₯ over two letters as pattern Sturmian if it had the lowest possible maximal pattern complexity, 2𝑛. Later, Kamae and Rao extended their definition of pattern Sturmian sequences to be sequences over β„“ β‰₯ 2 letters which are not periodic by projection and have maximal pattern complexity ℓ𝑛.

This dissertation answers a question posed by Kamae and Zamboni by characterizing recurrent pattern Sturmian sequences as either a coding of an irrational circle rotation by two intervals, or an element of a nearly simple Toeplitz subshift. We also show that nonrecurrent pattern Sturmian sequences are either very close to constant (such examples were given by Kamae and Zamboni) or a (nonrecurrent) coding of an irrational circle rotation by two intervals. We further examine the structure of sequences over more than two letters which have low maximal pattern complexity and provide a characterization for the structure of sequence over more than two letters with the lowest possible maximal pattern complexity.

Copyright Date

6-2026

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Casey Schlortt

Provenance

Received from ProQuest

File Format

application/pdf

Language

English (eng)

Extent

78 pgs

File Size

440 KB

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