Regularly renormalizable Hénon-like maps with bounded combinatorics have uniformly controlled stretching of horizontal curves, so their one-dimensional profiles are precompact.
Newhouse Laminations
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abstract
Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension $2$ laminations of maps with infinitely many sinks which move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: H\'enon maps, in any dimension, with infinitely many sinks, quadratic H\'enon-like maps with infinitely many sinks and a period doubling attractor, quadratic H\'enon-like maps with infinitely many sinks and a strange attractor, non trivial analytic families of polynomial maps with infinitely many sinks.
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A Priori Bounds for H\'enon-like Renormalization
Regularly renormalizable Hénon-like maps with bounded combinatorics have uniformly controlled stretching of horizontal curves, so their one-dimensional profiles are precompact.