On the multifractal spectrum of weighted Birkhoff averages

<p style='text-indent:20px;'>In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.</p>

Download Full-text

Glider automorphisms and a finitary Ryan’s theorem for transitive subshifts of finite type

Natural Computing ◽

10.1007/s11047-019-09759-1 ◽

2019 ◽

Vol 19 (4) ◽

pp. 773-786

Author(s):

Johan Kopra

Keyword(s):

Automorphism Group ◽

Finite Type ◽

Finite Collection ◽

Finite Point ◽

Element Subset ◽

Continuous Map ◽

Full Shift ◽

Subshifts Of Finite Type

Abstract For any mixing SFT X we construct a reversible shift-commuting continuous map (automorphism) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. As an application we prove a finitary Ryan’s theorem: the automorphism group $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) contains a two-element subset S whose centralizer consists only of shift maps. We also give an example which shows that a stronger finitary variant of Ryan’s theorem does not hold even for the binary full shift.

Download Full-text

Support stability of maximizing measures for shifts of finite type

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.69 ◽

2019 ◽

pp. 1-12

Author(s):

JULIANO S. GONSCHOROWSKI ◽

ANTHONY QUAS ◽

JASON SIEFKEN

Keyword(s):

Finite Type ◽

Fundamental Difference ◽

Probability Measures ◽

Two Dimensional ◽

Small Perturbations ◽

Subshift Of Finite Type ◽

Local Configuration ◽

Ergodic Optimization ◽

Full Shift ◽

Subshifts Of Finite Type

This paper establishes a fundamental difference between $\mathbb{Z}$ subshifts of finite type and $\mathbb{Z}^{2}$ subshifts of finite type in the context of ergodic optimization. Specifically, we consider a subshift of finite type $X$ as a subset of a full shift $F$ . We then introduce a natural penalty function $f$ , defined on $F$ , which is 0 if the local configuration near the origin is legal and $-1$ otherwise. We show that in the case of $\mathbb{Z}$ subshifts, for all sufficiently small perturbations, $g$ , of $f$ , the $g$ -maximizing invariant probability measures are supported on $X$ (that is, the set $X$ is stably maximized by $f$ ). However, in the two-dimensional case, we show that the well-known Robinson tiling fails to have this property: there exist arbitrarily small perturbations, $g$ , of $f$ for which the $g$ -maximizing invariant probability measures are supported on $F\setminus X$ .

Download Full-text

Transitive action on finite points of a full shift and a finitary Ryan’s theorem

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.84 ◽

2017 ◽

Vol 39 (06) ◽

pp. 1637-1667 ◽

Cited By ~ 2

Author(s):

VILLE SALO

Keyword(s):

Automorphism Group ◽

Commutator Subgroup ◽

Turing Machines ◽

Finite Support ◽

Transitive Action ◽

Group Invariant ◽

Finite Set ◽

Transitivity Property ◽

Full Shift ◽

Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s $V$ .

Download Full-text