Karl Petersen's Home Page

I am a faculty member (now retired) with the Mathematics Department of the University of North Carolina at Chapel Hill

Degrees: AB 1965 Princeton; MA 1967, Ph.D. 1969 Yale
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.

A couple of pictures.

Lecture Notes:

Information and Coding: Materials from First-Year Seminar

PDF version

Symbolic Dynamics--Math. 261, Spring 1998

PDF version

Attractors and Attracting Measures--Math. 261, Spring 1997

PDF version

Notes on dynamics of continued fractions from Spring 2000 Math. 261:

Postscript version

PDF version


Notes of lecture course at Workshop on Dynamics and Randomness, Santiago, Chile, December 2000, "Information Compression and Retention in Dynamical Processes":

Postscript version

PDF version


Short courses on ergodic theory in Egypt and Estonia

Postscript version

PDF version


Sofic Measures: Characterizations of hidden Markov chains by linear alebra, formal languages, and symbolic dynamics--Math. 210, Spring 2006

PDF version


Measure-Preserving Systems--Math. 857, Spring 2007

PDF version


Measuring Complexity in Cantor Dynamics---Lectures at the CIMPA Research School 2--13 November 2015

PDF version


B03 W03  D04 D05 C06

Recent Talks

Factor Maps on Shifts of Finite Type and Measures (or, Hidden Markov Chains Found Again). Eurandom, Technical University of Eindhoven, June 2005 and Ohio State University, April 2007.

Some Sturmian Symbolic Dynamics. Queen Mary University of London, June 22, 2009.

An adic dynamical system related to the Delannoy numbers. Workshop on Information and Randomness, Pucon, Chile, December 2010.

Some results and systems related to the super-K property. 2012 Workshop in Dynamical Systems and Related Topics, Pennsylvania State University.

Adic Systems and Symbolic Dynamics. Short Course at Pingree Park, Colorado, July 2014.

Generalizations of Neural Complexity to Dynamical Systems. University of Paris 6, Jussieu, June 16, 2015.


Slides of talks from Banff (BIRS) Workshop on Entropy Rate of Hidden Markov Processes and Connections to Dynamical Systems

Mike Boyle

Paul Cuff

Dongning Guo

Guangyue Han

Fred Juang

Erik Ordentlich

Henry Pfister

Mark Pollicott

Edgardo Ugalde

Evgeny Verbitskiy

Pascal Vontobel


Recent reprints and preprints:


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 TeX, so the TeX is not AMS-TeX or AMS-LaTeX, as for subsequent papers. Still one can obtain the content of the paper from this source even though it is not ideally formatted.)


Ergodic theorems and the basis of science
Synthese 108 (1996), 171-183.


Symmetric Gibbs measures (with Klaus Schmidt)
Trans. Amer. Math. Soc. 349 (1997), 2775-2811.


Binomial-coefficient multiples of irrationals (with Terrence M. Adams)
Monatsh. f. Math. 125 (1998), 269-278.


Factor maps between tiling dynamical systems
Forum Math. 11 (1999), 503-512.


Nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem
(with Michael Keane)


Measures of maximal relative entropy (with Anthony Quas and Sujin Shin)
Erg. Th. Dyn. Sys. 23 (2003), 207-223.

Tail fields generated by symbol counts in measure-preserving systems (with Jean-Paul Thouvenot)
Colloq. Math. 101 (2004), 9-23.


Dynamical properties of the Pascal adic transformation (with Xavier Méla)
Erg. Th. Dyn. Sys. 25 (2005), 227-256.


On the definition of relative pressure for factor maps on shifts of finite type (with Sujin Shin)
Bull. London Math. Soc. 37 (2005), 601-612.


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

PDF version


Basic Constructions and Examples (with Matthew Nicol), Springer Online Encyclopedia of Complexity

PDF version


Reinforced random walks and adic transformations (with Sarah Bailey Frick), J. Theoret. Probab. 23 (2010), no. 3, 920-943.

PDF version


The Euler adic dynamical system and path counts in the Euler graph (with Alexander Varchenko), Tokyo J. Math. 33, No. 2 (2010), 327-340.

PDF version


Path count asymptotics and Stirling numbers (with Alexander Varchenko), Proc. Amer. Math. Soc. 140 (2012), 1909-1919.

PDF version


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.

PDF version


An adic dynamical system related to the Delannoy numbers, Erg. Th. Dyn. Sys. 32 (2012), 809-823.

PDF version


Markov diagrams for some non-Markovian systems (with Kathleen Carroll), Contemp. Math. 678 (2016), 73-101.

PDF version


Dynamical properties of some adic systems with arbitrary orderings (with Sarah Frick and Sandi Shields), Erg. Th. Dyn. Sys. 2016.

PDF version


Dynamical intricacy and average sample complexity (with Benjamin Wilson), Dynamical Systems, http://dx.doi.org/10.1080/14689367.2017.1374352, 2017.

PDF version


Constructive symbolic presentations of rank one measure-preserving systems (with Terrence Adams and Sébastien Ferenczi), Colloq. Math. 150 (2017), 243-255.

PDF version


Tree shift complexity (with Ibrahim Salama), 2017.

PDF version




Brownian Motion, Hardy Spaces and Bounded Mean Oscillation , LMS Lecture Note Series 28, 1977.

Ergodic Theory, 1983; corrected paperback edition, 1989. Errata.

Ergodic Theory and Its Connections with Harmonic Analysis: Proceedings of the 1993 Alexandria Conference, with Ibrahim A. Salama, LMS Lecture Note Series 205, 1995.

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, 40 W. 20th St., New York, NY 10011-4211.



Calculus Explorer and Tutor I and II , 11 diskettes plus Student Guide, 1994.

HarperCollinsCollegePublishers, 1900 East Lake Ave., Glenview, IL 60025, ISBN 0-673-55834-7.