Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn structure of topological entropy for tree-shift of finite type
127
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
In this paper, we provide an effective method to compute the topological entropies of $G$-subshifts of finite type ($G$-SFTs) with $G=F_{d}$ and $S_{d}$, the free group and free semigroup with $d$ generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of $S_{2}$-SFTs, namely the types $mathbf{E},mathbf{D},mathbf{C}$ and $mathbf{O}$, are introduced and we could compute their entropies explicitly. This enables us to give the complete characterization of $S_{2}$-SFTs on two symbols. That is, the set of entropies of $S_{2}$-SFTs on two symbols is equal to $mathbf{E}cup mathbf{D}cup mathbf{C}cup mathbf{O}$. The methods developed in $S_{d}$-SFTs will also be applied to the study of the entropy theory of $F_{d}$-SFTs. The entropy formulae of $S_{d}$-, $F_{d}$-golden mean shifts and $k$-colored chessboards are also presented herein.
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.
This paper considers the topological degree of $G$-shifts of finite type for the case where $G$ is a nonabelian monoid. Whenever the Cayley graph of $G$ has a finite representation and the relationships among the generators of $G$ are determined by a matrix $A$, the coefficients of the characteristic polynomial of $A$ are revealed as the number of children of the graph. After introducing an algorithm for the computation of the degree, the degree spectrum, which is finite, relates to a collection of matrices in which the sum of each row of every matrix is bounded by the number of children of the graph. Furthermore, the algorithm extends to $G$ of finite free-followers.
We consider suspension flows with continuous roof function over the full shift $Sigma$ on a finite alphabet. For any positive entropy subshift of finite type $Y subset Sigma$, we explictly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over $Sigma$ are exactly the lifts of the measure(s) of maximal entropy for $Y$. In the case when $Y$ is transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If $Y$ has more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a Holder continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
