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### Open Access Journals

DCDS

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.

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.

JMD

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.

DCDS

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 $\lambda$).

DCDS

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.

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