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Register a new userInvariant Radon measures and minimal sets for subgroups of $text{Homeo}_+(mathbb{R})$
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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.
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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 $ngeq 2$.
We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was already studied by Choquet and Deny (1960) in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on broader class of systems satisfying the conditions: recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${mathbb R}$. Our work can be viewed as a subsequent paper of Babillot et al. (1997) and Deroin et al. (2013).
We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratners seminal work.
We consider the action of $SL(2,mathbb{R})$ on a vector bundle $mathbf{H}$ preserving an ergodic probability measure $ u$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $hat u$ is any lift of $ u$ to a probability measure on the projectivized bunde $mathbb{P}(mathbf{H})$ that is invariant under the upper triangular subgroup, then $hat u$ is supported in the projectivization $mathbb{P}(mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $mathbb{P}(mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $Sigma$, with hyperbolic foliation $mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension.
In this note, we give a nature action of the modular group on the ends of the infinite (p + 1)-cayley tree, for each prime p. We show that there is a unique invariant probability measure for each p.
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