Date of Award

8-1-2018

Document Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Ronnie Pavlov, Ph.D.

Second Advisor

Nicholas Ormes

Third Advisor

Alvaro Arias

Fourth Advisor

Brian Majestic

Keywords

Surface entropy, Entropy

Abstract

Let χ be the class of 1-D and 2-D subshifts. This thesis defines a new function, HS : χ x R → [0,∞] which we call the surface entropy of a shift. This definition is inspired by the topological entropy of a subshift and we compare and contrast several structural properties of surface entropy to entropy. We demonstrate that much like entropy, the finiteness of surface entropy is a conjugacy invariant and is a tool in the classification of subshifts. We develop a tiling algorithm related to continued fractions which allows us to prove a continuity result about surface entropy in the 2-D case, namely that while it is only upper semicontinuous with respect to eccentricity that there are bounds on how badly discontinuous it can behave.

A known result about entropy is that the class of entropies of 2-D SFTs is the class of CFA numbers. In the second part of this thesis we show that all such CFA numbers can be realized as the surface entropy of a 2-D SFT. Furthermore we construct an example of a 2-D SFT demonstrating that the class of surface entropies is a strict superset to the class of entropies.

Publication Statement

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

Provenance

Received from ProQuest

Rights holder

Dennis Pace

File size

104 p.

File format

application/pdf

Language

en

Discipline

Theoretical mathematics

Included in

Mathematics Commons

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