# Thomas K. French

## Teaching Assistant Professor

### What I do

I am a teaching assistant professor in the math department. I teach many courses including MATH 1200: Calculus for Business and Social Science as well as the calculus sequence, MATH 1951, 1952, and 1953.

### Specialization(s)

Symbolic Dynamics

### Professional Biography

I graduated with a bachelor's degree in math and chemistry from the University of Arkansas in 2009, then moved to Colorado to attend graduate school at the University of Denver, where I earned my master's in 2012 and Ph.D. in 2016. My thesis explored follower and extender sets of one-dimensional shift spaces, and my research interests continue to lie in the field of symbolic dynamics.

### Degree(s)

- Ph.D., Mathematics, University of Denver, 2016

### Research

A one-dimensional symbolic dynamical system is a pair (X, T), where X is a set of bi-infinite sequences of symbols from some finite alphabet A, T is the shift map, and X is closed and shift-invariant. My research has largely focused on follower, predecessor, and extender sets: given a word w, the follower set of w, denoted F(w), is the set of all right-infinite sequences u which may legally follow the word w. Similarly, the predecessor set P(w) is the set of all left-infinite sequences s which may legally precede w, and the extender set E(w) is the set of all pairs (s, u) where s is a left-infinite sequence, u is a right-infinite sequence, and swu is a legal point of X.

For each natural number n, we define the set F_X(n) to be the set of all distinct follower sets of words of length n. Then |F_X(n)| is the total number of distinct follower sets of words of length n in X. The follower set sequence of a shift space X is the sequence {|F_X(n)|}. (We define the predecessor set sequence and extender set sequence similarly.) Much of my research so far involves classifying what kinds of

sequences may be realized as follower, predecessor, or extender set sequences of shift spaces.

I have shown that for sofic shifts, a wide class of eventually periodic sequences may be realized as the shift's follower, predecessor, or extender set sequence, and that non-sofic shifts may have follower, predecessor, or extender set sequences which are not eventually monotone increasing. Using the classical beta-shifts as examples, I have shown that the follower, predecessor, and extender set sequences may exhibit vastly different growth rates. In joint work with Dr. Ronnie Pavlov, we have shown that the logarithmic growth rate of the extender set sequence (which we call extender entropy) is a conjugacy invariant.

For each natural number n, we define the set F_X(n) to be the set of all distinct follower sets of words of length n. Then |F_X(n)| is the total number of distinct follower sets of words of length n in X. The follower set sequence of a shift space X is the sequence {|F_X(n)|}. (We define the predecessor set sequence and extender set sequence similarly.) Much of my research so far involves classifying what kinds of

sequences may be realized as follower, predecessor, or extender set sequences of shift spaces.

I have shown that for sofic shifts, a wide class of eventually periodic sequences may be realized as the shift's follower, predecessor, or extender set sequence, and that non-sofic shifts may have follower, predecessor, or extender set sequences which are not eventually monotone increasing. Using the classical beta-shifts as examples, I have shown that the follower, predecessor, and extender set sequences may exhibit vastly different growth rates. In joint work with Dr. Ronnie Pavlov, we have shown that the logarithmic growth rate of the extender set sequence (which we call extender entropy) is a conjugacy invariant.

### Featured Publications

(2019). Follower, predecessor, and extender entropies. Monatshefte fur Mathematik, 188(3), 495-510.

. (2016). Subshifts with slowly growing numbers of follower sets. Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby. Boston, MA, USA: American Mathematical Society.

. (2016). Characterizing Follower and Extender Set Sequences. Dynamical Systems: An International Journal, 31(1), 293-310.

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