Generic family with robustly infinitely many sinks

By P. Berger. 2014

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We show, for every $$r>d\ge 0$$r>d≥0 or $$r=d\ge 2$$r=d≥2, the existence of a Baire generic set of $$C^d$$Cd-families of $$C^r$$Cr-maps $$(f_a)_{a\in {\mathbb {R}}^k}$$(fa)a∈Rk of a manifold M of dimension $$\ge $$≥2, so that for every a small the map $$f_a$$fa has infinitely many sinks. When the dimension of the manifold is $$\ge $$≥3, the generic set is formed by families of diffeomorphisms. When M is the annulus, this generic set is formed by local diffeomorphisms. This is a counter example to a conjecture of Pugh and Shub.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/599ff114f2d8b3e2ec560109fbaeea40aa6087cf