Karl Petersen's Home Page

I am a faculty member 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.

Office Hours: T,Th 11-12 and by appointment
Phone: 919-962-2380; E-mail: petersen@math.unc.edu

Teaching, Spring 2008

Mathematics 521, Advanced Calculus I
T-Th 9:30-10:45, Phillips 383

Syllabus

Writing up mathematics

Mathematics 56H, First Year Seminar, Information and Coding
(Fall 2007)

Some questions to think about

Syllabus

Computer setup

Writing up mathematics

Spreadsheets

Notes on Counting

Notes on Elementary Probability

Notes on Number Theory and Cryptography

Notes on Shannon's Information Theory

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

Lecture Notes:

Symbolic Dynamics--Math. 261, Spring 1998

Postscript version

PDF version

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

Postscript version

PDF version

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

Postscript version

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

Postscript version

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Short courses on ergodic theory in Egypt and Estonia

Postscript version

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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

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.

Euler and Reverse Euler Adics and Random Walks. Universities of Paris 6 and Rouen, May-June 2006.

Some Sturmian Symbolic Dynamics. Wesleyan Dynamics Conference in Honor of Ethan Coven, Oct. 13-14, 2007.

Reinforced Random Walks and Adic Transformations. Memphis, Banff, Paris, May-June 2008.

Recent reprints and preprints:

Random ergodic theorems with universally representative sequences (with Michael Lacey, Dan Rudolph, and Mate Wierdl)
Ann. Inst. H. Poincar� 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.
univ.dvi
univ.ps

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

Symmetric Gibbs measures (with Klaus Schmidt)
Trans. Amer. Math. Soc. 349 (1997), 2775-2811.
gibbs6.dvi
gibbs6.ps

Binomial-coefficient multiples of irrationals (with Terrence M. Adams)
Monatsh. f. Math. 125 (1998), 269-278.
adams8.dvi
adams8.ps

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

Nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem
(with Michael Keane)
KeanePetersenLNMS.dvi
KeanePetersenLNMS.pdf
KeanePetersenLNMS.ps

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

Tail fields generated by symbol counts in measure-preserving systems (with Jean-Paul Thouvenot)
Colloq. Math. 101 (2004), 9-23.
quant10.dvi
quant10.ps
quant10.pdf

Dynamical properties of the Pascal adic transformation (with Xavier M�la)
Erg. Th. Dyn. Sys. 25 (2005), 227-256.
part1.ps

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.
RPfinal.ps
RPfinal.pdf

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.
EulerErgodicity.pdf

Random permutations and unique fully supported ergodicity for the Euler adic transformation (with Sarah Bailey Frick), to appear in Ann. Inst. Henri Poincare Prob. Stat.
UEM.pdf

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

PDF version

Books:

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.

These are available from Cambridge University Press, 40 W. 20th St., New York, NY 10011-4211, phone 1-800-872-7423 in US and Canada. (The first one is probably out of print.)

Courseware:

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

HarperCollinsCollegePublishers, 1900 East Lake Ave., Glenview, IL 60025, phone 1-800-828-6000, e-mail harperglenvw@delphi.com. ISBN 0-673-55834-7.

STEVE KALIKOW's

Cover letter
Ergodic Theory book

SOME RECENT PICTURES