Solenoidal attractors with bounded combinatorics are shy

TL;DR

This paper proves that in generic families of multimodal maps, the parameters leading to solenoidal attractors with bounded combinatorics are extremely rare, occupying zero measure in the parameter space.

Contribution

It establishes that solenoidal attractors with bounded combinatorics are measure-zero phenomena in generic real-analytic multimodal maps.

Findings

Solenoidal attractors with bounded combinatorics are measure-zero in parameter space.

Generic families of multimodal maps rarely exhibit such attractors.

The result applies to real-analytic maps in finite dimensions.

Abstract

We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lebesgue measure.