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.

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.

scholarly journals Structure of transition classes for factor codes on shifts of finite type

2014 ◽  
Vol 35 (8) ◽  
pp. 2353-2370 ◽  
Author(s):  
MAHSA ALLAHBAKHSHI ◽  
SOONJO HONG ◽  
UIJIN JUNG
Keyword(s):  
Finite Type ◽  
Irreducible Factor ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Transitive Point ◽  
Dynamical Properties ◽  
Factor Code

Given a factor code ${\it\pi}$ from a shift of finite type $X$ onto a sofic shift $Y$, the class degree of ${\it\pi}$ is defined to be the minimal number of transition classes over the points of $Y$. In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.

scholarly journals Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation

2019 ◽  
Vol 40 (9) ◽  
pp. 2552-2570
Author(s):  
SCOTT SCHMIEDING
Keyword(s):  
Asymptotic Behavior ◽  
Automorphism Group ◽  
Finite Type ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Shift Of Finite Type ◽  
Dimension Group ◽  
Bounded Below

Let $(X_{A},\unicode[STIX]{x1D70E}_{A})$ be a shift of finite type and $\text{Aut}(\unicode[STIX]{x1D70E}_{A})$ its corresponding automorphism group. Associated to $\unicode[STIX]{x1D719}\in \text{Aut}(\unicode[STIX]{x1D70E}_{A})$ are certain Lyapunov exponents $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$, which describe asymptotic behavior of the sequence of coding ranges of $\unicode[STIX]{x1D719}^{n}$. We give lower bounds on $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$ in terms of the spectral radius of the corresponding action of $\unicode[STIX]{x1D719}$ on the dimension group associated to $(X_{A},\unicode[STIX]{x1D70E}_{A})$. We also give lower bounds on the topological entropy $h_{\text{top}}(\unicode[STIX]{x1D719})$ in terms of a distinguished part of the spectrum of the action of $\unicode[STIX]{x1D719}$ on the dimension group, but show that, in general, $h_{\text{top}}(\unicode[STIX]{x1D719})$ is not bounded below by the logarithm of the spectral radius of the action of $\unicode[STIX]{x1D719}$ on the dimension group.

scholarly journals Relative equilibrium states and class degree

2017 ◽  
Vol 39 (4) ◽  
pp. 865-888
Author(s):  
MAHSA ALLAHBAKHSHI ◽  
JOHN ANTONIOLI ◽  
JISANG YOO
Keyword(s):  
Finite Type ◽  
Upper Bound ◽  
Relative Equilibrium ◽  
Ergodic Measure ◽  
Equilibrium States ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Ergodic Measures ◽  
Factor Code ◽  
Special Case

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$, and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$. If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$. This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.

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.

scholarly journals The automorphism group of a shift of finite type

1988 ◽  
Vol 306 (1) ◽  
pp. 71-71 ◽  
Author(s):  
Mike Boyle ◽  
Douglas Lind ◽  
Daniel Rudolph
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Shift Of Finite Type

scholarly journals Glider automata on all transitive sofic shifts

2021 ◽  
pp. 1-29
Author(s):  
JOHAN KOPRA
Keyword(s):  
Automorphism Group ◽  
Cellular Automaton ◽  
Finite Collection ◽  
Finite Point ◽  
Element Subset ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Reversible Cellular Automaton

Abstract For any infinite transitive sofic shift X we construct a reversible cellular automaton (that is, an automorphism of the shift X) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible cellular automaton which is sensitive with respect to all directions. As another application we prove a finitary version of Ryan’s theorem: the automorphism group $\operatorname {\mathrm {Aut}}(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of S-gap shifts these results do not extend beyond the sofic case.

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 Topologically completely positive entropy and zero-dimensional topologically completely positive entropy

2017 ◽  
Vol 38 (5) ◽  
pp. 1894-1922
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Finite Type ◽  
Sufficient Condition ◽  
Positive Entropy ◽  
Two Dimensional ◽  
Completely Positive ◽  
Shift Of Finite Type ◽  
One Dimensional ◽  
Zero Entropy ◽  
The Difference

In a previous paper [Pavlov, A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys.34 (2014), 2054–2065], the author gave a characterization for when a $\mathbb{Z}^{d}$-shift of finite type has no non-trivial subshift factors with zero entropy, a property which we here call zero-dimensional topologically completely positive entropy. In this work, we study the difference between this notion and the more classical topologically completely positive entropy of Blanchard. We show that there are one-dimensional subshifts and two-dimensional shifts of finite type which have zero-dimensional topologically completely positive entropy but not topologically completely positive entropy. In addition, we show that strengthening the hypotheses of the main result of Pavlov [A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys.34 (2014), 2054–2065] yields a sufficient condition for a $\mathbb{Z}^{d}$-shift of finite type to have topologically completely positive entropy.

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