Karl Petersen

Dynamical properties of the Pascal adic transformation

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascals triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.

Chains, entropy, coding

Various definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic – they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.

Measures of maximal relative entropy

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 projects to ν and has maximal entropy among all measures in the fiber, but there is an explicit bound on the number of such maximal entropy preimages.

Random permutations and unique fully supported ergodicity for the Euler adic transformation

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 partially justify a frequent assumption about the equidistribution of random permutations.

Ergodicity of the adic transformation on the Euler graph

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 the Eulerian number A(n,k), the number of permutations of 1,2,...,n+1 with exactly n-k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.

Information Compression and Retention in Dynamical Processes

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 s-shifts), and factor mappings between symbolic or tiling dynamical systems.

Entropy of Hidden Markov Processes and Connections to Dynamical Systems: Papers from the Banff International Research Station Workshop

Hidden Markov processes (HMPs) are important objects of study in many areas of pure and applied mathematics, including information theory, probability theory, dynamical systems and statistical physics, with applications in electrical engineering, computer science and molecular biology. This collection of research and survey papers presents important new results and open problems, serving as a unifying gateway for researchers in these areas. Based on talks given at the Banff International Research Station Workshop, 2007, this volume addresses a central problem of the subject: computation of the Shannon entropy rate of an HMP. This is a key quantity in statistical physics and information theory, characterizing the fundamental limit on compression and closely related to channel capacity, the limit on reliable communication. Also discussed, from a symbolic dynamics and thermodynamical viewpoint, is the problem of characterizing the mappings between dynamical systems which map Markov measures to Markov (or Gibbs) measures, and which allow for Markov lifts of Markov chains.

On the definition of relative pressure for factor maps on shifts of finite type

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 to every invariant measure. With a suitable extension of one of the definitions, the same holds true for any continuous potential function.

The helical transform as a connection between ergodic theory and harmonic analysis

Direct proofs are given for the formal equivalence of the L 2 -boundedness of the maximal operators corresponding to the partial sums of Fourier series, the range of a discrete helical walk, partial Fourier coefficients, and the discrete helical transform. Strong (2, 2) for the double maximal (ergodic) helical transforms is extended to actions of R d and Z d . It is also noted that the spectral measure of a measure-preserving flow has a continuity property at ∞, the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak (1, 1)

SOME CONNECTIONS BETWEEN ERGODIC THEORY AND HARMONIC ANALYSIS

Publisher Summary Parallels, connections, and cross-applications among ergodic theory, harmonic analysis, and spectral theory have long been observed and exploited. This chapter focuses on a family of operators that exist in the boundary zone of these three areas, the study of which clarifies some of these interelationships: the rotated ergodic Hubert transform, or the helical transform of an integrable function f on a measure space X with respect to a measure-preserving transformation T. There is an analogous class of operators for a measure-preserving flow {Tt: −∞ < t < ∞}. The chapter discusses the relationships among strong (p, p) inequalities for the maximal helical transform and its variants, and the suprema of the partial sums of Fourier series.