Hidden Markov processes in the context of symbolic dynamics (2009)
In an effort to aid communication among different fields and perhaps facilitate progress on problems common to all of them, this article discusses hidden Markov processes from several viewpoints,...
TAIL FIELDS GENERATED BY SYMBOL COUNTS IN MEASURE-PRESERVING SYSTEMS (2008)
Karl Petersen, Jean-paul Thouvenot
Abstract. A finite-state stationary process is called (one or twosided) super-K if its (one or two-sided) super-tail field—generated by keeping track of (initial or central) symbol counts as well...
NOTES ON SHANNON’S INFORMATION THEORY (2008)
We will learn here (1) that entropy is a measure of the fundamental information content of messages, giving the minimal number of binary bits per symbol needed to encode the source; (2) that...
Karl Petersen, Rika Hagihara, Jessica Hubbs, Nathan Pennington, Yuki Yayama
In symbolic dynamics and the theory of stationary processes, Bernoulli and Markov measures are among the first examples of measures examined. They have been thoroughly studied and many of their...
ON THE DEFINITION OF RELATIVE PRESSURE FOR FACTOR MAPS ON SHIFTS OF FINITE TYPE (2008)
Abstract. We show that two natural definitions of the relative pressure function for a locally constant potential function and a factor map from a shift of finite type coincide almost everywhere with...
Abstract: We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem. Let (X, B, µ) be a probability space, T: X → X a...
Sarah Bailey Frick, Karl Petersen
permutations and unique fully supported ergodicity for the Euler adic transformation
NOTES ON ELEMENTARY PROBABILITY (2008)
Probability theory is an attempt to work mathematically with the relative uncertainties of random events. In order to get started, we do not attempt to estimate the probability of occurrence of any...
Abstract: We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem. Let (X, B, µ) be a probability space, T: X → X a...
Karl Petersen, Matthew Bonzek, Joshua Clemons, Jeanette Olli, Andres Del Junco
Abstract. These notes are from a graduate course given in Spring 2007 at the University of North Carolina at Chapel Hill. A major portion will be published as part of the Springer Online Encyclopedia...
Measures of maximal relative entropy (2008)
Karl Petersen, Anthony Quas, Sujin Shin
Abstract. Given an irreducible subshift of finite type X, a subshift Y, a factor map π: X → Y, and an ergodic invariant measure ν on Y, there can exist more than one ergodic measure on X which...
Cristóbal Rojas González, Stefano Galatolo, Cristian Calude, Serge Troubetzkoy, Gilles Dowek, Stefano Galatolo, ...
Aléatoire et théorie ergodique: un point
ERGODIC THEOREMS AND THE BASIS OF SCIENCE (2007)
Abstract. New results in ergodic theory show that averages of repeated measurements will typically diverge with probability one if there are random errors in the measurement of time. Since...
Michael Lacey, Karl Petersen, Dan Rudolph, Or Z
ABSTRACT. When elements of a measure-preserving action of R d
26> g, Z ff ?g (f \Gamma ) = 1 0: Assume first that f 2 L 1 . Fix N = 1; 2; : : : , and let EN = ff N ? g: Notice that (f \Gamma )ØEN (f \Gamma ); since x = 2 EN implies (f \Gamma )(x) 0. Thus...
A Comment on the Definition of Relative Pressure (2007)
We show that two natural de nitions of the relative pressure function for a locally constant potential function and a factor map from a shift of nite type coincide almost everywhere with respect to...
Notes On Number Theory And (2007)
Cryptography Karl Petersen, Karl Petersen, Karl Petersen
ements. We will see in a moment that it is possible to do some arithmetic with these elements. Think of the integers as being written on a long ribbon, which is then wrapped around a circle of...
TAIL FIELDS GENERATED BY SYMBOL COUNTS IN MEASURE-PRESERVING SYSTEMS (2007)
Karl Petersen, Jean-paul Thouvenot
Abstract. A nite-state stationary process is called (one or twosided) super-K if its (one or two-sided) super-tail eld|generated by keeping track of (initial or central) symbol counts as well as of...
Dynamical properties of the Pascal adic transformation, Erg (2007)
Abstract. We study the dynamics of a transformation that acts on innite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return...
Random permutations and unique fully supported ergodicity for the Euler adic transformation (2007)
Frick, Sarah Bailey, Petersen, Karl
There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may...
Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem (2006)
Keane, Michael, Petersen, Karl
We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.
Ergodicity of the adic transformation on the Euler graph (2006)
Bailey, Sarah, Keane, Michael, Petersen, Karl, Salama, Ibrahim
The Euler graph has vertices labelled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is...
Ergodicity of the adic transformation on the Euler graph (2006)
Sarah Bailey, Michael Keane, Karl Petersen, Ibrahim A. Salama, S. Bailey, M. Keane, ...
The Euler graph has vertices labelled (n,k) for n = 0,1,2,... and k = 0,1,...,n, with k + 1 edges from (n,k) to (n + 1,k) and n − k + 1 edges from (n,k) to (n + 1,k + 1). The number of paths from...
A comment on the definition of relative pressure (2004)
We show that two natural definitions of the relative pressure function for a locally constant potential function and a factor map from a shift of finite type coincide almost everywhere with respect...
Then � {f ∗ (f − λ) ≥ 0. (2004)
Michael Keane, Karl Petersen, Michael Keane, Karl Petersen
Let (X, B, µ) be a probability space, T: X → X a (possibly noninvertible) measure-preserving transformation, and f ∈ L 1 (X, B, µ). Let Akf = 1 k−1 � k j=0 fT j, f ∗ N = sup 1≤k≤N...
Dynamical properties of the Pascal adic transformation (2003)
We study the dynamics of a transformation that acts on infinite pathsin the graph associated with Pascal's triangle. For each ergodicinvariant measure the asymptotic law of the return time to...
Dynamical properties of the Pascal adic transformation (2003)
We study the dynamics of a transformation that acts on infinite pathsin the graph associated with Pascal's triangle. For each ergodicinvariant measure the asymptotic law of the return time to...
Dynamical properties of the Pascal adic transformation (2003)
We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to...
Dynamical properties of the Pascal adic transformation (2003)
We study the dynamics of a transformation that acts on infinite pathsin the graph associated with Pascal's triangle. For each ergodicinvariant measure the asymptotic law of the return time to...
Dynamical properties of the Pascal adic transformation (2003)
We study the dynamics of a transformation that acts on infinite pathsin the graph associated with Pascal's triangle. For each ergodicinvariant measure the asymptotic law of the return time to...
Measures of maximal relative entropy (2002)
Petersen, Karl, Quas, Anthony, Shin, Sujin
Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi : X \to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects...
Information compression and retention in dynamical processes (2001)
We discuss some recent work on various constructions that accumulate or remove information within dynamical systems: tail fields, numeration systems and formal languages (especially of beta-shifts),...
Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem (2000)
A very short and direct proof along the lines of the Kamae-Katznelson-Weiss approach.
Factor Maps Between Tiling Dynamical Systems (1999)
. We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such systems which cannot be achieved by working within a finite...
Factor maps between tiling dynamical systems (1998)
We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such systems which cannot be achieved by working within a finite...
Symmetric Gibbs measures (1997)
Abstract. We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic...
Symmetric Gibbs measures (1996)
Petersen, Karl, Schmidt, Klaus
We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the...
Symmetric Gibbs Measures (1995)
. We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation...
Cyclical pumping--a separation of an air and carbon dioxide mixture. (1969)
Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1969.
Eine modifizierte Ziehl-Neelsen-Färbung: Jodtinktur als Mittel zur Nachfärbung. (1935)
Kiel, Med. Diss., 1937.