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Register a new userTopological Entropy for Shifts of Finite Type Over $mathbb{Z}$ and Tree
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We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and $h(mathcal{T}_X) geq h(X)$, where $mathcal{T}_X$ is the hom tree-shift derived from $X$. We characterize a necessary and sufficient condition when the equality holds for the case where $X$ is a shift of finite type. In addition, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, which makes such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type is equal to or larger than that of its maximal irreducible component.
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This paper deals with the topological entropy for hom Markov shifts $mathcal{T}_M$ on $d$-tree. If $M$ is a reducible adjacency matrix with $q$ irreducible components $M_1, cdots, M_q$, we show that $h(mathcal{T}_{M})=max_{1leq ileq q}h(mathcal{T}_{M_{i}})$ fails generally, and present a case study with full characterization in terms of the equality. Though that it is likely the sets ${h(mathcal{T}_{M}):Mtext{ is binary and irreducible}}$ and ${h(mathcal{T}_{X}):Xtext{ is a one-sided shift}}$ are not coincident, we show the two sets share the common closure. Despite the fact that such closure is proved to contain the interval $[d log 2, infty)$, numerical experiments suggest its complement contain open intervals.
We reveal an algorithm for determining the complete prefix code irreducibility (CPC-irreducibility) of dyadic trees labeled by a finite alphabet. By introducing an extended directed graph representation of tree shift of finite type (TSFT), we show that the CPC-irreducibility of TSFTs is related to the connectivity of its graph representation, which is a similar result to one-dimensional shifts of finite type.
The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift $X$ and its associated hom tree-shift $mathcal{T}_{X}$, as well as the equivalence in the sofic shift. On the other hand, we investigate the interrelationship among the comparable mixing properties on tree-shifts as those on multidimensional shift spaces. They include irreducibility, topologically mixing, block gluing, and strong irreducibility, all of which are defined in the spirit of classical multidimensional shift, complete prefix code (CPC), and uniform CPC. In summary, the mixing properties defined in all three manners coincide for $mathcal{T}_{X}$. Furthermore, an equivalence between irreducibility on $mathcal{T}_{A}$ and irreducibility on $X_A$ are seen, and so is one between topologically mixing on $mathcal{T}_{A}$ and mixing property on $X_A$, where $X_A$ is the one-sided shift space induced by the matrix $A$ and $T_A$ is the associated tree-shift. These equivalences are consistent with the mixing properties on $X$ or $X_A$ when viewed as a degenerate tree-shift.
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $mathbb{Z}^k$-actions which are generated by random compositions of the generators of $mathbb{Z}^k$-actions. Applying Pesins theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $mathbb{Z}^k$(or $mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and $C^{1}$ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{Z}^k$-actions.
Let $(X, T)$ be a topological dynamical system (TDS), and $h (T, K)$ the topological entropy of a subset $K$ of $X$. $(X, T)$ is {it lowerable} if for each $0le hle h (T, X)$ there is a non-empty compact subset with entropy $h$; is {it hereditarily lowerable} if each non-empty compact subset is lowerable; is {it hereditarily uniformly lowerable} if for each non-empty compact subset $K$ and each $0le hle h (T, K)$ there is a non-empty compact subset $K_hsubseteq K$ with $h (T, K_h)= h$ and $K_h$ has at most one limit point. It is shown that each TDS with finite entropy is lowerable, and that a TDS $(X, T)$ is hereditarily uniformly lowerable if and only if it is asymptotically $h$-expansive.
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