# [Todos CMAT] Seminario y despedida

Rafael Potrie rpotrie en cmat.edu.uy
Vie Jul 19 10:53:46 -03 2019

Hola,

Hoy a las 14.30 escucharemos a Braulio, que se está yendo mañana luego de
pasar un año aquí. Abajo título y resumen.

sds

Topological and rotational aspects of homoclinic bifurcation in the annulus

In this talk, we will discuss some aspects of the dynamics of annulus
homeomorphisms  $f:\mathbb{A}\rightarrow\mathbb{A}$

which has an attracting closed annular region $A\subset \mathbb{A}$

(i.e. $A$ is homeomorphic to $\mathbb{S}^1\times[-1,1]$ and
$f(A)\subset \textrm{Int}(A)$). In this situation, an $f$-invariant
set $$\mathcal{A}_f=\bigcap_{n\geq 0}f^n(A)$$ exist and is an
essential annular continuum

(a compact connected set that separates the annulus into

exactly two components $U^{\pm}(\mathcal{A}_f)$).

The topology of such continua can be very intricate (for instance,
they can be hairy", indecomposable, or even hereditarily
indecomposable).

Let $\rho(f,\mathcal{A}_f)$ be the rotation set in $\mathcal{A}_f$.

By a theorem of Poincaré on circle homeomorphisms,

if the attractor $\mathcal{A}_f$ is homeomorphic to $\mathbb{S}^1$,

then $\rho(f,\mathcal{A}_f)$ is a singleton.

On the other hand, Barge and Gillette [92] prove that any attracting
circloid $\mathcal{A}_f$  (i.e. it contains no proper essential
annular subcontinua) with empty interior and rotation set
non-degenerate must be indecomposable.

Concrete examples are the Birkhoff attractors for certain class of
dissipative twist maps in the annulus (Le Calvez [90]).

>From some recent works, it is known that if the attractor is a
circloid and $\rho(f,\mathcal{A}_f)$ is a

non trivial rotation interval, then the dynamic has complexity;

for instance there are infinitely many periodic points of arbitrarily
large periods,

uncountably many ergodic measures and positive topological entropy.

We will present the results concerning the change of the attractor and
the rotation set

$\rho(f,\mathcal{A}_f)$ in terms of $C^r$-perturbations of $f$,

for $r\geq0$, whose $f$ belongs to a class of diffeomorphisms on the
annulus with a homoclinic tangency associated to a dissipative
hyperbolic fixed point $P$ whose an unstable manifold is dense in
$\mathcal{A}_f$.

In addition, we will discuss the continuity of the prime end rotation
numbers $\rho^{\pm}(\mathcal{A}_f)$,

associated to their complementary regions $U^{\pm}(\mathcal{A}_f)$,
restricted to a family.

--
Rafael Potrie
rafaelpotrie en gmail.com
http://www.cmat.edu.uy/~rpotrie/
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