We study attracting invariant graphs and attracting multi-graphs for skew product systems. First, we focus on skew product systems driven by a baker map having an n-dimensional closed ball $B_n \subseteq \mathbb{R}^n$ as the fiber and investigate the geometrical structures of invariant graphs and multi-graphs for these systems. We introduce an n-porcupine attractor which is a generalization of porcupine horseshoes discovered by Diaz and Gelfert. We construct an open set in the space of all such skew products such that any skew product belonging to this set admits a non-uniformly hyperbolic maximal attractor. Morover this attractor either is an n-porcupine attractor or an attracting continuous invariant graph. Then, we provide some related results on the ergodic properties of attracting graphs and investigate stability results for such graphs under deterministic perturbations. In our context the rates of contraction are non-uniform thus we have non-uniformly hyperbolic attractors that are the support of ergodic SRB measures. Additionally, we prove that the SRB measure varies continuously with the skew product in the Hutchinson metric. Furthermore, we construct robust attracting multi-graphs or porcupine multi-graphs for skew products driven by a baker map. Finally, for a certain class of skew products driven by expanding circle maps we show that the following dichotomy is ascertained: the non-uniformly maximal attractor is either a massive attractor or a thick attractor.

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt