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Register a new userThe nonexistence of expansive actions on Suslinian continua by groups of subexponential growth
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We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive.
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Let $X$ be a non-degenerate connected compact metric space. If $X$ admits a distal minimal action by a finitely generated amenable group, then the first vCech cohomology group $ {check H}^1(X)$ with integer coefficients is nontrivial. In particular, if $X$ is homotopically equivalent to a CW complex, then $X$ cannot be simply connected.
A dynamical system is a pair $(X,G)$, where $X$ is a compact metrizable space and $G$ is a countable group acting by homeomorphisms of $X$. An endomorphism of $(X,G)$ is a continuous selfmap of $X$ which commutes with the action of $G$. One says that a dynamical system $(X,G)$ is surjunctive provided that every injective endomorphism of $(X,G)$ is surjective (and therefore is a homeomorphism). We show that when $G$ is sofic, every expansive dynamical system $(X,G)$ with nonnegative sofic topological entropy and satisfying the weak specification and the strong topological Markov properties, is surjunctive.
By the {em Suslinian number} $Sln(X)$ of a continuum $X$ we understand the smallest cardinal number $kappa$ such that $X$ contains no disjoint family $C$ of non-degenerate subcontinua of size $|C|gekappa$. For a compact space $X$, $Sln(X)$ is the smallest Suslinian number of a continuum which contains a homeomorphic copy of $X$. Our principal result asserts that each compact space $X$ has weight $leSln(X)^+$ and is the limit of an inverse well-ordered spectrum of length $le Sln(X)^+$, consisting of compacta with weight $leSln(X)$ and monotone bonding maps. Moreover, $w(X)leSln(X)$ if no $Sln(X)^+$-Suslin tree exists. This implies that under the Suslin Hypothesis all Suslinian continua are metrizable, which answers a question of cite{DNTTT1}. On the other hand, the negation of the Suslin Hypothesis is equivalent to the existence of a hereditarily separable non-metrizable Suslinian continuum. If $X$ is a continuum with $Sln(X)<2^{aleph_0}$, then $X$ is 1-dimensional, has rim-weight $leSln(X)$ and weight $w(X)geSln(X)$. Our main tool is the inequality $w(X)leSln(X)cdot w(f(X))$ holding for any light map $f:Xto Y$.
Denote by $DC(M)_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^infty$ manifold $M$. We show that if $dim(M)geq2$ and $r eq dim(M)+1$, then any homomorphism from $DC(M)_0$ to ${Diff}^1(R)$ or ${Diff}^1(S^1)$ is trivial.
Let $BS(1,n) =< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ ngeq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $. This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid. We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $<f,h | h circ f circ h^{-1} = f^n>$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $TT^2$. When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.
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