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.

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 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 On the computability of rotation sets and their entropies

2018 ◽  
Vol 40 (2) ◽  
pp. 367-401 ◽  
Author(s):  
MICHAEL A. BURR ◽  
MARTIN SCHMOLL ◽  
CHRISTIAN WOLF
Keyword(s):  
Dynamical System ◽  
Finite Type ◽  
Topological Entropy ◽  
Metric Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Subshift Of Finite Type ◽  
Continuous Potential ◽  
Rotation Set ◽  
Entropy Functions

Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.

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.

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.

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.

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.

scholarly journals Approximating the maximum ergodic average via periodic orbits

2008 ◽  
Vol 28 (4) ◽  
pp. 1081-1090 ◽  
Author(s):  
D. COLLIER ◽  
I. D. MORRIS
Keyword(s):  
Invariant Measure ◽  
Finite Type ◽  
Periodic Orbits ◽  
Ergodic Average ◽  
Probability Measures ◽  
Hölder Continuous ◽  
Subshift Of Finite Type ◽  
Stretched Exponential ◽  
Rate Of Approximation ◽  
Borel Probability

AbstractLet σ:ΣA→ΣA be a subshift of finite type, let $\mathcal {M}_\sigma $ be the set of all σ-invariant Borel probability measures on ΣA, and let $f\colon \Sigma _A \to \mathbb {R}$ be a Hölder continuous observable. There exists at least one σ-invariant measure μ which maximizes $\int f \,d\mu $. The following question was asked by B. R. Hunt, E. Ott and G. Yuan: how quickly can the maximum of the integrals $\int f \,d\mu $ be approximated by averages along periodic orbits of period less than p? We give an example of a Hölder observable f for which this rate of approximation is slower than stretched-exponential in p.

Non-uniqueness of measures of maximal entropy for subshifts of finite type

1994 ◽  
Vol 14 (2) ◽  
pp. 213-235 ◽  
Author(s):  
Robert Burton ◽  
Jeffrey E. Steif
Keyword(s):  
Finite Type ◽  
Higher Dimensions ◽  
One Dimension ◽  
Maximal Entropy ◽  
Subshift Of Finite Type ◽  
Unique Measure ◽  
Extremal Measures ◽  
Subshifts Of Finite Type ◽  
Measures Of Maximal Entropy ◽  
Measure Of Maximal Entropy

AbstractIt is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.

scholarly journals Subsystems, Perron numbers, and continuous homomorphisms of Bernoulli shifts

1989 ◽  
Vol 9 (3) ◽  
pp. 561-570 ◽  
Author(s):  
Selim Tuncel
Keyword(s):  
Finite Type ◽  
Bernoulli Shift ◽  
Block Code ◽  
Block Codes ◽  
Probability Vector ◽  
Finiteness Conditions ◽  
Subshift Of Finite Type ◽  
Bernoulli Shifts ◽  
Subshifts Of Finite Type ◽  
Markov Measures

AbstractLet S, T be subshifts of finite type, with Markov measures p, q on them, and let φ: (S, p) → (T, q) be a block code. Let Ip, Iq denote the information cocycles of p, q. For a subshift of finite type ⊂T, the pressure of equals that of . Applying this to Bernoulli shifts and using finiteness conditions on Perron numbers, we have the following. If the probability vector p = (p1…, pk+1) is such that the distinct transcendental elements of {p1/pk+1…pk/pk+1) are algebraically independent then the Bernoulli shift B(p) has finitely many Bernoulli images by block codes.

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