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Register a new userThe topological structure of function space of transitive maps
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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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Let $G$, $R$ and $A$ be topological groups. Suppose that $G$ and $R$ act continuously on $A$, and $G$ acts continuously on $R$. In this paper, we define a partially crossed topological $G-R$-bimodule $(A,mu)$, where $mu:Arightarrow R$ is a continuous homomorphism. Let $Der_{c}(G,(A,mu))$ be the set of all $(alpha,r)$ such that $alpha:Grightarrow A$ is a continuous crossed homomorphism and $mualpha(g)=r^{g}r^{-1}$. We introduce a topology on $Der_{c}(G,(A,mu))$. We show that $Der_{c}(G,(A,mu))$ is a topological group, wherever $G$ and $R$ are locally compact. We define the first cohomology, $H^{1}(G,(A,mu))$, of $G$ with coefficients in $(A,mu)$ as a quotient space of $Der_{c}(G,(A,mu))$. Also, we state conditions under which $H^{1}(G,(A,mu))$ is a topological group. Finally, we show that under what conditions $H^{1}(G,(A,mu))$ is one of the following: $k$-space, discrete, locally compact and compact.
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.
We consider the topological behaviors of continuous maps with one topological attractor on compact metric space $X$. This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. We provide a leveled $A$-$R$ pair decomposition for such maps, and characterize $alpha$-limit set of each point. Based on weak Morse decomposition of $X$, we construct a bounded Lyapunov function $V(x)$, which give a clear description of orbit behavior of each point in $X$ except a meager set.
We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we investigate whether they have various properties, including topological transitivity, topological mixing, dense periodic points, and specification. To each Markov multi-map, we associate a shift of finite type (SFT), and then our main results relate the properties of the SFT with those of the Markov multi-map. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.
Let $text{Homeo}_{+}(mathbb{S}^1)$ denote the group of orientation preserving homeomorphisms of the circle $mathbb{S}^1$. A subgroup $G$ of $text{Homeo}_{+}(mathbb{S}^1)$ is tightly transitive if it is topologically transitive and no subgroup $H$ of $G$ with $[G: H]=infty$ has this property; is almost minimal if it has at most countably many nontransitive points. In the paper, we determine all the topological conjugation classes of tightly transitive and almost minimal subgroups of $text{Homeo}_{+}(mathbb{S}^1)$ which are isomorphic to $mathbb{Z}^n$ for any integer $ngeq 2$.
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