# Thomas 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

### Degree(s)

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

### Research

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.