scholarly journals Lower entropy factors of sofic systems

1983 ◽  
Vol 3 (4) ◽  
pp. 541-557 ◽  
Author(s):  
Mike Boyle
Keyword(s):  
Periodic Point ◽  
Finite Type ◽  
Embedding Theorem ◽  
Subshift Of Finite Type ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Continuous Maps ◽  
Lower Entropy ◽  
Subshifts Of Finite Type ◽  
General Method

AbstractA mixing subshift of finite type T is a factor of a sofic shift S of greater entropy if and only if the period of any periodic point of S is divisible by the period of some periodic point of T. Mixing sofic shifts T satisfying this theorem are characterized, as are those mixing sofic shifts for which Krieger's Embedding Theorem holds. These and other results rest on a general method for extending shift-commuting continuous maps into mixing subshifts of finite type.

scholarly journals Automorphisms of one-sided subshifts of finite type

1990 ◽  
Vol 10 (3) ◽  
pp. 421-449 ◽  
Author(s):  
Mike Boyle ◽  
John Franks ◽  
Bruce Kitchens
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Finite Order ◽  
Subshift Of Finite Type ◽  
Finite Subgroups ◽  
Subshifts Of Finite Type

AbstractWe prove that the automorphism group of a one-sided subshift of finite type is generated by elements of finite order. For one-sided full shifts we characterize the finite subgroups of the automorphism group. For one-sided subshifts of finite type we show that there are strong restrictions on the finite subgroups of the automorphism group.

scholarly journals Normal amenable subgroups of the automorphism group of sofic shifts

2020 ◽  
pp. 1-14
Author(s):  
KITTY YANG
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Amenable Subgroup

Let $(X,\unicode[STIX]{x1D70E})$ be a transitive sofic shift and let $\operatorname{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\operatorname{Aut}(X)$ must be contained in the subgroup generated by the shift. We also show that the result does not extend to higher dimensions by giving an example of a two-dimensional mixing shift of finite type due to Hochman whose automorphism group is amenable and not generated by the shift maps.

scholarly journals Embedding subshifts of finite type into the Fibonacci–Dyck shift

2016 ◽  
Vol 37 (6) ◽  
pp. 1862-1886
Author(s):  
TOSHIHIRO HAMACHI ◽  
WOLFGANG KRIEGER
Keyword(s):  
Finite Type ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Subshift Of Finite Type ◽  
Subshifts Of Finite Type ◽  
Necessary And Sufficient

A necessary and sufficient condition is given for the existence of an embedding of an irreducible subshift of finite type into the Fibonacci–Dyck shift.

Lattice invariants for sofic shifts

1991 ◽  
Vol 11 (4) ◽  
pp. 787-801 ◽  
Author(s):  
Susan Williams
Keyword(s):  
Finite Type ◽  
Necessary Conditions ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Dimension Group ◽  
Sofic Shifts ◽  
Factor Map ◽  
Factor Maps

AbstractTo a factor map φ from an irreducible shift of finite type ΣAto a sofic shiftS, we associate a subgroup of the dimension group (GA, Â) which is an invariant of eventual conjugacy for φ. This invariant yields new necessary conditions for the existence of factor maps between equal entropy sofic shifts.

Uniform Sampling of Subshifts of Finite Type on Grids and Trees

2017 ◽  
Vol 28 (03) ◽  
pp. 263-287 ◽  
Author(s):  
Jean Mairesse ◽  
Irène Marcovici
Keyword(s):  
Finite Type ◽  
Nearest Neighbor ◽  
Sampling Methods ◽  
Probability Measures ◽  
Probabilistic Cellular Automata ◽  
Subshift Of Finite Type ◽  
Uniform Sampling ◽  
Subshifts Of Finite Type ◽  
Uniform Random Sampling ◽  
Insight Into

Let us color the vertices of the grid ℤd or the infinite regular tree 𝕋d, using a finite number of colors, with the constraint that some predefined pairs of colors are not allowed for adjacent vertices. The set of admissible colorings is called a nearest-neighbor subshift of finite type (SFT). We study “uniform” probability measures on SFT, with the motivation of having an insight into “typical” admissible configurations. We recall the known results on uniform measures on SFT on grids and we complete the picture by presenting some contributions to the description of uniform measures on SFT on 𝕋d. Then we focus on the problem of uniform random sampling of configurations of SFT. We propose a first method based on probabilistic cellular automata, which is valid under some restrictive conditions. Then we concentrate on the case of SFT on ℤ for which we propose several alternative sampling methods.

scholarly journals Decompositions of factor codes and embeddings between shift spaces with unequal entropies

2012 ◽  
Vol 33 (1) ◽  
pp. 144-157
Author(s):  
SOONJO HONG ◽  
UIJIN JUNG ◽  
IN-JE LEE
Keyword(s):  
Finite Type ◽  
Shift Spaces ◽  
Sofic Shift ◽  
Shift Space ◽  
Sofic Shifts ◽  
The Given ◽  
Factor Code

AbstractGiven a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.

Support stability of maximizing measures for shifts of finite type

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 $F\setminus X$ .

scholarly journals Topological entropy and periodic points of a factor of a subshift of finite type

1986 ◽  
Vol 104 ◽  
pp. 117-127 ◽  
Author(s):  
Takashi Shimomura
Keyword(s):  
Finite Type ◽  
Topological Entropy ◽  
Compact Space ◽  
Metric Spaces ◽  
Periodic Points ◽  
Subshift Of Finite Type ◽  
Continuous Map ◽  
Continuous Maps ◽  
Definition Of ◽  
Uniformly Continuous

Let X be a compact space and f be a continuous map from X into itself. The topological entropy of f, h(f), was defined by Adler, Konheim and McAndrew [1]. After that Bowen [4] defined the topological entropy for uniformly continuous maps of metric spaces, and proved that the two entropies coincide when the spaces are compact. The definition of Bowen is useful in calculating entropy of continuous maps.

scholarly journals Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

2010 ◽  
Vol 31 (4) ◽  
pp. 1109-1161 ◽  
Author(s):  
J.-R. CHAZOTTES ◽  
J.-M. GAMBAUDO ◽  
E. UGALDE
Keyword(s):  
Finite Type ◽  
Recursive Algorithm ◽  
Temperature Limit ◽  
Finite Union ◽  
Subshift Of Finite Type ◽  
Frobenius Theorem ◽  
The Family ◽  
Finite Set ◽  
Constant Potential ◽  
Subshifts Of Finite Type

AbstractLet A be a finite set and let ϕ:Aℤ→ℝ be a locally constant potential. For each β>0 (‘inverse temperature’), there is a unique Gibbs measure μβϕ. We prove that as β→+∞, the family (μβϕ)β>0 converges (in the weak-* topology) to a measure that we characterize. This measure is concentrated on a certain subshift of finite type, which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron–Frobenius theorem for matrices à la Birkhoff. The crucial idea we bring is a ‘renormalization’ procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.

Textile systems and one-sided resolving automorphisms and endomorphisms of the shift

2008 ◽  
Vol 28 (1) ◽  
pp. 167-209 ◽  
Author(s):  
MASAKAZU NASU
Keyword(s):  
Finite Type ◽  
Structure Theory ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
Subshifts Of Finite Type

AbstractTwo results on textile systems are obtained. Using these we prove that for any automorphism φ of any topologically-transitive subshift of finite type, if φ is expansive and φ or φ−1 has memory zero or anticipation zero, then φ is topologically conjugate to a subshift of finite type. Moreover, this is generalized to a result on chain recurrent onto endomorphisms of topologically-transitive subshifts of finite type. Using textile systems and textile subsystems, we develop a structure theory concerning expansiveness with the pseudo orbit tracing property on directionally essentially weakly one-sided resolving automorphisms and endomorphisms of subshifts.

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