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Register a new userLowering topological entropy over subsets
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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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Let $(X, T)$ be a topological dynamical system. Denote by $h (T, K)$ and $h^B (T, K)$ the covering entropy and dimensional entropy of $Ksubseteq X$, respectively. $(X, T)$ is called D-{it lowerable} (resp. {it lowerable}) if for each $0le hle h (T, X)$ there is a subset (resp. closed subset) $K_h$ with $h^B (T, K_h)= h$ (resp. $h (T, K_h)= h$); is called D-{it hereditarily lowerable} (resp. {it hereditarily lowerable}) if each Souslin subset (resp. closed subset) is D-lowerable (resp. lowerable). In this paper it is proved that each topological dynamical system is not only lowerable but also D-lowerable, and each asymptotically $h$-expansive system is D-hereditarily lowerable. A minimal system which is lowerable and not hereditarily lowerable is demonstrated.
In this note a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are: the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowens entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for $1leq pleqinfty$ the Bowen $p$-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen $infty$-entropy for compact metric spaces coincide.
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.
We prove the entropy conjecture of M. Barge from 1989: for every $rin [0,infty]$ there exists a pseudo-arc homeomorphism $h$, whose topological entropy is $r$. Until now all pseudo-arc homeomorphisms with known entropy have had entropy $0$ or $infty$.
Let $C(mathbf I)$ be the set of all continuous self-maps from ${mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $fin C({mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in $mathbf{I}$, there exists a positive integer $n$ such that $Ucap f^{-n}(V) ot=emptyset.$ We note $T(mathbf{I})$ and $overline{T(mathbf{I})}$ to be the sets of all transitive maps and its closure in the space $C(mathbf I)$. In this paper, we show that $T(mathbf{I})$ and $overline{T(mathbf{I})}$ are homeomorphic to the separable Hilbert space $ell_2$.
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