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.

Topological entropy of Markov set-valued functions

2019 ◽  
Vol 41 (2) ◽  
pp. 321-337 ◽  
Author(s):  
LORI ALVIN ◽  
JAMES P. KELLY
Keyword(s):  
Finite Type ◽  
Topological Entropy ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Interval Function ◽  
Shift Of Finite Type ◽  
Shift Space ◽  
Interval Functions

We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

Perturbations of multidimensional shifts of finite type

2010 ◽  
Vol 31 (2) ◽  
pp. 483-526 ◽  
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.

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 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 METASTABILITY, LYAPUNOV EXPONENTS, ESCAPE RATES, AND TOPOLOGICAL ENTROPY IN RANDOM DYNAMICAL SYSTEMS

2013 ◽  
Vol 13 (04) ◽  
pp. 1350004 ◽  
Author(s):  
GARY FROYLAND ◽  
OGNJEN STANCEVIC
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Topological Entropy ◽  
Lower Bounds ◽  
Random Systems ◽  
Random Dynamical Systems ◽  
Upper And Lower Bounds ◽  
Random System ◽  
Random Maps ◽  
Frobenius Operator

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

Common extensions and hyperbolic factor maps for coded systems

1995 ◽  
Vol 15 (3) ◽  
pp. 517-534 ◽  
Author(s):  
Doris Fiebig
Keyword(s):  
Dynamical Systems ◽  
Finite Type ◽  
Topological Entropy ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Coded Systems ◽  
Factor Maps

AbstractThe classification of dynamical systems by the existence of certain common extensions has been carried out very successfully in the class of shifts of finite type (‘finite equivalence’, ‘almost topological conjugacy‘). We consider generalizations of these notions in the class of coded systems. Topological entropy is shown to be a complete invariant for the existence of a common coded entropy preserving extension. In contrast to the shift of finite type setting, this extension cannot be made bounded-to-1 in general. Common extensions with hyperbolic factor maps lead to a version of almost topological conjugacy for coded systems.

A Note on Permutations and Topological Entropy of Continuous Maps of the Interval

1986 ◽  
Vol 29 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Bill Byers
Keyword(s):  
Periodic Orbit ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Continuous Maps ◽  
Bounded Below

AbstractSuppose f is a continuous endomorphism of an interval which has a periodic orbit, p0 < P1 < … < pn, that defines a permutation a by f(pi) = pσ(i). If σ is irreducible the topological entropy of f is bounded below by the logarithm of the spectral radius of an n x n matrix which is induced by σ.

scholarly journals On structure of topological entropy for tree-shift of finite type

2021 ◽  
Vol 292 ◽  
pp. 325-353
Author(s):  
Jung-Chao Ban ◽  
Chih-Hung Chang ◽  
Wen-Guei Hu ◽  
Yu-Liang Wu
Keyword(s):  
Finite Type ◽  
Topological Entropy ◽  
Shift Of Finite Type

Resolving factor maps for shifts of finite type with equal entropy

1991 ◽  
Vol 11 (2) ◽  
pp. 219-240 ◽  
Author(s):  
Jonathan Ashley
Keyword(s):  
Periodic Point ◽  
Finite Type ◽  
Shift Of Finite Type ◽  
Dimension Group ◽  
Almost Everywhere ◽  
Point Condition ◽  
Factor Maps ◽  
Commuting Map

AbstractWe sharpen a result of Boyle, Marcus and Trow as follows. An aperiodic shift of finite type ΣAfactors onto another ΣBwith equal entropy by a 1-to-l almost everywhere right-closing map if and only if (1) the dimension group for ΣBis a quotient of that for ΣA; and (2) ΣAand ΣBsatisfy the trivial periodic point condition for existence of a shift-commuting map from ΣAto ΣB.

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