Periodic points in random substitution subshifts
- PDF / 385,071 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 91 Downloads / 166 Views
Periodic points in random substitution subshifts Dan Rust1 Received: 12 June 2019 / Accepted: 14 August 2020 © The Author(s) 2020
Abstract We study various aspects of periodic points for random substitution subshifts. In order to do so, we introduce a new property for random substitutions called the disjoint images condition. We provide a procedure for determining the property for compatible random substitutions—random substitutions for which a well-defined abelianisation exists. We find some simple necessary criteria for primitive, compatible random substitutions to admit periodic points in their subshifts. In the case that the random substitution further has disjoint images and is of constant length, we provide a stronger criterion. A method is outlined for enumerating periodic points of any specified length in a random substitution subshift. Keywords Random substitutions · Periodic points · Topological entropy Mathematics Subject Classification 37B10 · 37A50 · 37B40 · 52C23 Random substitutions are a generalisation of the classical notion of a substitution on a finite alphabet. In the classical setting, letters are mapped to words over the same alphabet, and then this map is iterated. Dynamical systems associated with these classical substitutions are well studied and there is a large community devoted to solving some of the few remaining big problems in this area [1,5,8]. In the setting of random substitutions, letters have a set of possible words (often with an accompanying probability distribution) to which they may be independently mapped. There, one must contend with all possible outcomes of iteration, where each letter of a word is mapped independently of all others. This leads to an exponential growth in the number
Communicated by H. Bruin.
B 1
Dan Rust [email protected] Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
123
D. Rust
of words admitted in the language of a random substitution and a corresponding explosion of complexity for the associated subshift of bi-infinite sequences over that language. Accordingly, the dynamical systems and tilings associated with random substitutions provide good models for quasicrystaline structures that have long range order induced by an underlying hierarchical supertile structure, whilst also possessing positive entropy. Such models are highly sought-after in the world of solid-state physics [9] and have also proved useful for the study of molecular evolution [13,16] where so-called expansion-modification systems are a model proposed to explain long-range correlations of sequences associated with DNA. The recent study of random substitutions has lead to rapid advances in our understanding of various topological, dynamical and diffractive properties of their associated tilings and subshifts [2,3,11,15,23]. With this greater understanding has come a multitude of simple to state but non-trivial open problems, some of which were outlined in recent articles [11,23]. The purpose of this article is to tackle one par
Data Loading...
