Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

<p style='text-indent:20px;'>In this text I study the asymptotics of the complexity function of <i>minimal</i> multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [<xref ref-type="bibr" rid="b12">12</xref>] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the <inline-formula><tex-math id="M1">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M2">\begin{document}$ [0, d] $\end{document}</tex-math></inline-formula>. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all <inline-formula><tex-math id="M3">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula> the possible values for entropy dimensions of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional SFT are the <inline-formula><tex-math id="M5">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M6">\begin{document}$ [0, d-1] $\end{document}</tex-math></inline-formula>. In the present text I prove formally this result for <inline-formula><tex-math id="M7">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case <inline-formula><tex-math id="M8">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> requires some substantial changes to be made in order to adapt the construction that are not discussed here.</p>

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Sofic subshifts and piecewise isometric systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385799151964 ◽

1999 ◽

Vol 19 (6) ◽

pp. 1485-1501 ◽

Cited By ~ 18

Author(s):

AREK GOETZ

Keyword(s):

Symbolic Dynamics ◽

Regular Partition ◽

Zero Entropy ◽

Point Set ◽

Subshifts Of Finite Type ◽

Piecewise Continuous ◽

Symbolic Coding ◽

Group Of Isometries ◽

Regular Partitions ◽

The Relationship

We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

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Medvedev degrees of two-dimensional subshifts of finite type

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.152 ◽

2012 ◽

Vol 34 (2) ◽

pp. 679-688 ◽

Cited By ~ 11

Author(s):

STEPHEN G. SIMPSON

Keyword(s):

Finite Type ◽

Equivalence Class ◽

Symbolic Dynamics ◽

Recursion Theory ◽

Two Dimensional ◽

Infinite Collection ◽

Subshifts Of Finite Type ◽

Medvedev Degrees

AbstractIn this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial computable functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices ℰs and ℰw of Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰs and ℰwconsist precisely of the Medvedev degrees and Muchnik degrees of two-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of two-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007859 ◽

1994 ◽

Vol 14 (2) ◽

pp. 213-235 ◽

Cited By ~ 49

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.

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Optimal state amalgamation is NP-hard

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.105 ◽

2017 ◽

Vol 39 (7) ◽

pp. 1857-1869 ◽

Cited By ~ 1

Author(s):

RAFAEL M. FRONGILLO

Keyword(s):

Finite Type ◽

Directed Graph ◽

Symbolic Dynamics ◽

Hitting Set ◽

State Splitting ◽

Continuous Maps ◽

Np Complete ◽

Subshifts Of Finite Type ◽

Hitting Set Problem ◽

Inverse Operation

A state amalgamation of a directed graph is a node contraction which is only permitted under certain configurations of incident edges. In symbolic dynamics, state amalgamation and its inverse operation, state splitting, play a fundamental role in the theory of subshifts of finite type (SFT): any conjugacy between SFTs, given as vertex shifts, can be expressed as a sequence of symbol splittings followed by a sequence of symbol amalgamations. It is not known whether determining conjugacy between SFTs is decidable. We focus on conjugacy via amalgamations alone and consider the simpler problem of deciding whether one can perform $k$ consecutive amalgamations from a given graph. This problem also arises when using symbolic dynamics to study continuous maps, where one seeks to coarsen a Markov partition in order to simplify it. We show that this state amalgamation problem is NP-complete by reduction from the hitting set problem, thus giving further evidence that classifying SFTs up to conjugacy may be undecidable.

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Perturbations of multidimensional shifts of finite type

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000040 ◽

2010 ◽

Vol 31 (2) ◽

pp. 483-526 ◽

Cited By ~ 4

Author(s):

RONNIE PAVLOV

Keyword(s):

Finite Type ◽

Lower Bounds ◽

Symbolic Dynamics ◽

Upper And Lower Bounds ◽

Shift Of Finite Type ◽

Strongly Irreducible ◽

Dimensional Cube

AbstractIn this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

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