Wild attractors for Fibonacci maps

TL;DR

This paper investigates the conditions under which Fibonacci maps in one-dimensional dynamics have wild attractors, providing a computable criterion and computer-assisted proofs to delineate parameter ranges for their existence.

Contribution

It introduces a constructive trichotomy for the Lebesgue measure of Fibonacci attractors and uses computer-assisted methods to identify parameter thresholds for wild attractor existence.

Findings

Fibonacci maps with critical degree d=3.8 lack wild attractors.

Fibonacci maps with critical degree d=5.1 have wild attractors.

A computable criterion determines the measure scenarios for Fibonacci attractors.

Abstract

Existence of wild attractors -- attractors whose basin has a positive Lebesgue measure but is not a residual set -- has been one of central themes in one-dimensional dynamics. It has been demonstrated by H. Bruin et al. that Fibonacci maps with a sufficiently flat critical point admit a wild attractor. We propose a constructive trichotomy that describes possible scenarios for the Lebesgue measure of the Fibonacci attractor based on a computable criterion. We use this criterion, together with a computer-assisted proof of existence of a Fibonacci renormalization $2$-cycle for non-integer critical degrees, to demonstrate that Fibonacci maps do not have a wild attractor when the degree of the critical point is $d=3.8$ (and, conjecturally, for $2< d \le 3.8$), and do admit it when $d=5.1$ (and, conjecturally, for $d \ge 5.1$).