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Register a new userTopological conjugation classes of tightly transitive subgroups of $text{Homeo}_{+}(mathbb{S}^1)$
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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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Let $G$ be a subgroup of $text{Homeo}_+(mathbb{R})$ without crossed elements. We show the equivalence among three items: (1) existence of $G$-invariant Radon measures on $mathbb R$; (2) existence of minimal closed subsets of $mathbb R$; (3) nonexistence of infinite towers covering the whole line. For a nilpotent subgroup $G$ of $text{Homeo}_+(mathbb{R})$, we show that $G$ always has an invariant Radon measure and a minimal closed set if every element of $G$ is $C^{1+alpha} (alpha>0$); a counterexample of $C^1$ commutative subgroup of $text{Homeo}_+(mathbb{R})$ is constructed.
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 $ell_2$.
Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $mathbb {P}^n_mathbb C$ preserving the unit ball $mathbb {H}^n_mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $mathbb P^n_{mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $mathbb {P}^n_{mathbb C}$ which are tangent to $partial mathbb {H}^n_mathbb {C}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(G )$, a closed $G$-invariant subset of $partial mathbb {H}^n_mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.
In this report, we first recall the Poincares classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys classification theorem for minimal orientation-preserving group actions on the circle. Then we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.
The kink Casimir effect in the massive non-linear $S^3$-sigma model is analyzed.
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