Degrees: AB 1965
Specialty: Ergodic theory
Ergodic theory is a fairly new branch of mathematics which applies probability and analysis to study the long-term average behavior of complicated systems. It overlaps heavily with (smooth) dynamical systems theory and draws methods, examples, and problems from harmonic analysis, number theory, combinatorics, and many other branches of mathematics. Applications range from celestial mechanics through interactions of biological populations to the efficient transmission and recording of information. My particular research interests have concentrated on symbolic dynamics, almost everywhere convergence, maximal theorems, and connections of ergodic theory with harmonic analysis and probability.
Information and Coding: Materials from First-Year Seminar
Symbolic Dynamics--Math. 261, Spring 1998
Attractors and Attracting Measures--Math. 261, Spring 1997
Notes on dynamics of continued fractions from Spring 2000 Math. 261:
Notes of lecture course at Workshop on Dynamics and
Short courses on ergodic theory in
Sofic Measures: Characterizations of hidden Markov chains by linear alebra, formal languages, and symbolic dynamics--Math. 210, Spring 2006
Measure-Preserving Systems--Math. 857, Spring 2007
Measuring Complexity in Cantor Dynamics---Lectures at the CIMPA Research School 2--13 November 2015
Random ergodic theorems with universally
representative sequences (with Michael Lacey, Dan Rudolph, and Mate Wierdl)
Ann. Inst. H. Poincaré 30 (1994), 353-395.
(The form here is an earlier version, produced by Exp to
Measures of maximal relative entropy (with
Anthony Quas and Sujin Shin)
Erg. Th. Dyn. Sys. 23 (2003), 207-223.
Dynamical properties of the Pascal adic
transformation (with Xavier Méla)
Erg. Th. Dyn. Sys. 25 (2005), 227-256.
Ergodicity of the adic transformation on the Euler
graph (with Sarah Bailey, Michael Keane, and Ibrahim Salama), Math. Proc. Camb. Phil. Soc. 141 (2006), 231-238.
Random permutations and unique fully supported
ergodicity for the Euler adic transformation (with Sarah Bailey Frick),
Ann. Inst. Henri Poincare Prob. Stat. 44 (2008), 876-885.
Measure-Preserving Systems, Springer Online Encyclopedia of Complexity
Basic Constructions and Examples (with Matthew Nicol), Springer Online Encyclopedia of Complexity
Reinforced random walks and adic transformations (with Sarah Bailey Frick), J. Theoret. Probab. 23 (2010), no. 3, 920-943.
The Euler adic dynamical system and path counts in the Euler graph (with Alexander Varchenko), Tokyo J. Math. 33, No. 2 (2010), 327-340.
Path count asymptotics and Stirling numbers (with Alexander Varchenko), Proc. Amer. Math. Soc. 140 (2012), 1909-1919.
Hidden Markov processes in the context of symbolic dynamics (with Mike Boyle), in Entropy of Hidden Markov Processes and Connections to Dynamical Systems, London Math. Soc. Lecture Note Ser., 385, Cambridge Univ. Press, Cambridge, 2011, 5-71.
An adic dynamical system related to the Delannoy numbers, Erg. Th. Dyn. Sys. 32 (2012), 809-823.
Markov diagrams for some non-Markovian systems (with Kathleen Carroll), Contemp. Math. 678 (2016), 73-101.
Dynamical properties of some adic systems with arbitrary orderings (with Sarah Frick and Sandi Shields), Erg. Th. Dyn. Sys. 2016.
Dynamical intricacy and average sample complexity (with Benjamin Wilson), Dynamical Systems, http://dx.doi.org/10.1080/14689367.2017.1374352, 2017.
Constructive symbolic presentations of rank one measure-preserving systems (with Terrence Adams and Sébastien Ferenczi), Colloq. Math. 150 (2017), 243-255.
Tree shift complexity (with Ibrahim Salama), 2017.
Brownian Motion, Hardy Spaces and Bounded Mean Oscillation , LMS Lecture Note Series 28, 1977.
Ergodic Theory and Its Connections with Harmonic
Analysis: Proceedings of the 1993
Entropy of Hidden Markov Processes and Connections to Dynamical Systems: Papers from the Banff International Research Station Workshop, with Brian Marcus and Tsachy Weissman, LMS Lecture Note Series 385, 2011.
These are available from Cambridge University Press,
Calculus Explorer and Tutor I and II , 11 diskettes plus Student Guide, 1994.