Actions of Baumslag-Solitar groups on surfaces
Nancy Guelman Isabelle Liousse
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 1945-1964 doi: 10.3934/dcds.2013.33.1945
Let $BS(1, n) =< a, b \ | \ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 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 $\mathbb{T} ^2$, 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 o f o 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 $S= \mathbb{T}^2$, 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 $\mathbb{T}^2$.
    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 $C^1$ and isotopic to identity then $fix(f)$ contains any minimal set.
keywords: Baumslag Solitar group Actions on surfaces minimal sets.
Axiom A diffeomorphisms derived from Anosov flows
Christian Bonatti Nancy Guelman
Journal of Modern Dynamics 2010, 4(1): 1-63 doi: 10.3934/jmd.2010.4.1
Let $M$ be a closed $3$-manifold, and let $X_t$ be a transitive Anosov flow. We construct a diffeomorphism of the form $f(p)=Y_{t(p)}(p)$, where $Y$ is an Anosov flow equivalent to $X$. The diffeomorphism $f$ is structurally stable (satisfies Axiom A and the strong transversality condition); the non-wandering set of $f$ is the union of a transitive attractor and a transitive repeller; and $f$ is also partially hyperbolic (the direction $\RR.Y$ is the central bundle).
keywords: partial hyperbolicity AxiomA diffeomorphism Birkhoff sections Anosov flows perturbations.
Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements
Juan Alonso Nancy Guelman Juliana Xavier
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1817-1827 doi: 10.3934/dcds.2015.35.1817
Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces of genus $g\geq 1$ with (pseudo)-Anosov elements. That is, we consider a closed surface $S$ of genus $g\geq 1$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is (pseudo)-Anosov with stretch factor $\lambda >1$. We show that $\langle f,h \rangle$ is not a faithful representation of $BS(1, n)$ if $\lambda > n$. We also show that there are no faithful representations of $BS(1, n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $\lambda$).
keywords: Rigidity theory. Baumslag-Solitar groups group actions pseudo-Anosov homeomorphisms Surface homeomorphisms
Examples of minimal set for IFSs
Nancy Guelman Jorge Iglesias Aldo Portela
Discrete & Continuous Dynamical Systems - A 2017, 37(10): 5253-5269 doi: 10.3934/dcds.2017227

We exhibit different examples of minimal sets for an IFS of homeomorphisms with rational rotation number. It is proved that these examples are, from a topological point of view, the unique possible cases.

keywords: Iterated function systems minimal sets

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