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.