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Karl Petersen

Publications and source records attributed to Karl Petersen.

At least 19 recordsLinked to original sources
arXiv2024

Polynomial shape adic systems are inherently expansive

To study any dynamical system it is useful to find a partition that allows essentially faithful encoding (injective, up to a small exceptional set) into a subshift. Most topological and measure-theoretic systems can be represented by Bratteli-Vershik (or adic, or BV) systems. So it is natural to ask when can a BV system be encoded essentially faithfully. We show here that for BV diagrams defined by homogeneous positive integer multivariable polynomials, and a wide family of their generalizations, which we call polynomial shape diagrams, for every choice of the edge ordering the coding according to initial path segments of a fixed finite length is injective off of a negligible exceptional set.

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arXiv2024

Arrival of information at a target set in a network

We consider labelings of a finite regular tree by a finite alphabet subject to restrictions specified by a nonnegative transition matrix, propose an algorithm for determining whether the set of possible configurations on the last row of the tree is independent of the symbol at the root, and prove that the algorithm succeeds in a bounded number of steps, provided that the dimension of the tree is greater than or equal to the maximum row sum of the transition matrix. (The question was motivated by calculation of topological pressure on trees and is an extension of the idea of primitivity for nonnegative matrices.)

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arXiv2024

Density of group languages in shift spaces

The density of a rational language can be understood as the frequency of some pattern in the shift space, for example a pattern like "words with an even number of a given letter." We study the density of group languages, i.e. rational languages recognized by morphisms onto finite groups, inside shift spaces. We show that the density with respect to any given ergodic measure on a shift space exists for every group language, because it can be computed by using any ergodic lift of the given measure to a skew product between the shift space and the recognizing group. We then further study densities in shifts of finite type (with a suitable notion of irreducibility), and then in minimal shifts. In the latter case, we obtain a closed formula for the density under the condition that the aforementioned skew product has minimal closed invariant subsets which are ergodic under the product of the original measure and the uniform probability measure on the group. The formula is derived in part from a characterization of minimal closed invariant subsets for skew products between shifts and finite groups relying on notions of cocycles and coboundaries. In the case where the whole skew product is ergodic under the product measure, then the density is just the cardinality of the subset of the group which defines the language divided by the cardinality of the group. Moreover, we provide sufficient conditions for the skew product to have minimal closed invariant subsets that are ergodic under the product measure. Finally, we investigate the link between minimal closed invariant subsets, return words and bifix codes.

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arXiv2022

Asymptotic pressure on some self-similar trees

The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix, respecting nearest neighbor adjacency restrictions. To the set of these configurations one can associate a pressure, which is defined as the limit, when it exists, of averages of the logarithm of the partition function over certain finite subgraphs. We prove that for shifts of finite type on generalized Fibonacci trees, under an added condition, the limit exists and is given by an infinite series. We also show that the limit of any cluster points of the pressure on finite subtrees as the number of generators grows without bound, which we call the asymptotic pressure, equals the logarithm of the maximum row sum of the interaction matrix.

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arXiv2022

A classification of nonexpansive Bratteli-Vershik systems

We study simple, properly ordered nonexpansive Bratteli-Vershik ($BV$) systems. Correcting a mistake in an earlier paper, we redefine the classes standard nonexpansive ($SNE$) and strong standard nonexpansive ($SSNE$). We define also the classes of very well timed and well timed systems, their opposing classes of untimed and very untimed systems (which feature, as subclasses of "Case (2)", in the work of Downarowicz and Maass as well as Hoynes on expansiveness of $BV$ systems of finite topological rank), and several related classes according to the existence of indistinguishable pairs (of some "depth") and their synchronization ("common cuts"). We establish some properties of these types of systems and some relations among them. We provide several relevant examples, including a problematic one that is conjugate to a well timed system while also (vacuously) in the classes "Case (2)". We prove that the class of all simple, properly ordered nonexpansive $BV$ systems is the disjoint union of the ones conjugate to well timed systems and those conjugate to untimed systems, thereby showing that nonexpansiveness in $BV$ systems arises in one of two mutually exclusive ways.

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arXiv2020

Periodic codings of Bratteli-Vershik systems

We develop conditions for the coding of a Bratteli-Vershik system according to initial path segments to be periodic, equivalently for a constructive symbolic recursive scheme corresponding to a cutting and stacking process to produce a periodic sequence. This is a step toward understanding when a Bratteli-Vershik system can be essentially faithfully represented by means of a natural coding as a subshift on a finite alphabet.

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arXiv2019

Entropy on regular trees

We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is $\mathbb Z$. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on $\mathbb Z^2$, we show that the entropy of the hard square tree shift on the regular $k$-tree increases with $k$, in contrast to the case of $\mathbb Z^k$. We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for $k=2,\dots,8$ and propose that this holds for all $k \geq 2$. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the $k$-tree for all $k$. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.

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arXiv2018

Tree shift topological entropy

We give a definition of topological entropy for tree shifts, prove that the limit in the definition exists, and show that it dominates the topological entropy of the associated one-dimensional shift of finite type when the labeling of the tree shares the same restrictions.

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arXiv2016

Dynamical intricacy and average sample complexity

We propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define \emph{intricacy} and \emph{average sample complexity} for topological and measure-preserving dynamical systems. We establish basic properties of these quantities, show that their suprema over covers or partitions equal the ordinary entropies, compute them for many shifts of finite type, and indicate natural directions for further research.

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arXiv2016

Constructive symbolic presentations of rank one measure-preserving systems

Given a rank one measure-preserving system defined by cutting and stacking with spacers, we produce a rank one binary sequence such that its orbit closure under the shift transformation, with its unique {nonatomic} invariant probability, is isomorphic to the given system. In particular, the classical dyadic odometer is presented in terms of a recursive sequence of blocks on the two-symbol alphabet $\{0,1\}$. The construction is accomplished using a definition of rank one in the setting of adic, or Bratteli-Vershik, systems.

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arXiv2016

Measuring Complexity in Cantor Dynamics

These notes from lectures by the author at the CIMPA Research School held in Salta, Argentina, in November 2015 review definitions and properties of various measures of complexity in topological and measure-preserving dynamical systems, especially symbolic dynamical systems.

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arXiv2015

Dynamical properties of some adic systems with arbitrary orderings

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli-Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every $k$ the subshift that arises from coding orbits according to the first $k$ edges is topologically weakly mixing. We give a necessary and sufficient condition for any adic system to be topologically conjugate to an odometer and use this condition to determine the probability that a random order on a fixed diagram, or a diagram constructed at random in some way, is topologically conjugate to an odometer. We also show that the closure of the union over all orderings of the subshifts arising from codings of the Pascal adic by the first edge has superpolynomial complexity, is not topologically transitive, and has no periodic points besides the two fixed points, while the intersection over all orderings consists of just four orbits.

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arXiv2015

Markov Diagrams for Some Non-Markovian Systems

Markov diagrams provide a way to understand the structures of topological dynamical systems. We examine the construction of such diagrams for subshifts, including some which do not have any nontrivial Markovian part, in particular Sturmian systems and some substitution systems.

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arXiv2011

An adic dynamical system related to the Delannoy numbers

We introduce an adic (Bratteli-Vershik) dynamical system based on a diagram whose path counts from the root are the Delannoy numbers. We identify the ergodic invariant measures, prove total ergodicity for each of them, and initiate the study of the dimension group and other dynamical properties.

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arXiv2010

Hidden Markov processes in the context of symbolic dynamics

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, especially that of symbolic dynamics, where they are known as sofic measures, or continuous shift-commuting images of Markov measures. It provides background, describes known tools and methods, surveys some of the literature, and proposes several open problems.

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arXiv2006

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.

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