Uniformly Expanding Full Branch Maps with a Wild Attracting Point

TL;DR

The paper demonstrates that uniformly expanding full-branch interval maps can possess a wild fixed-point attractor with a basin of full Lebesgue measure, challenging the intuition that strong expansion precludes wild attractors. It develops a framework with a countable partition accumulating at the fixed point, introduces invariant sets $C$ and $E$ built from cylinder structure, and uses convex preimage estimates to establish a positive lower bound on a left-cylindrical basin that evolves into full measure under stronger convexity assumptions. Through a nested-sets argument and a crucial bound on the residual set via the quantity $P=\prod p_i$, the authors prove $|B_0|=1$ for maps in $\mathcal{F}_*^{strong}$, and provide an explicit constructive example of such a map. Additionally, they discuss a nonexpanding variant when the construction parameter satisfies $c\ge2$, illustrating a dichotomy in expansion strength while highlighting the robustness of the wild attractor mechanism. These results expand the landscape of ergodic theory for interval maps by showing that wild attractors can coexist with strong, even infinite-entropy, expansion.

Abstract

We give a counterintuitive example of a uniformly expanding full branch map such that the point zero is a wild attractor.

Figures (4)

• Figure 1: Example of a map in $\mathcal{F}_{*}^{strong}$.
• Figure 2: Nested sets $I \supset C_1 \supset C_2$.
• Figure 3: Visual Proof of Lemma \ref{['Lem_Convex']}
• Figure 4: A map $f$ satisfying Proposition \ref{['Prop_Existence']}, with $c=1.5$.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 3.1
• proof
• proof : Proof of Proposition \ref{['Prop_Meagre']}
• Lemma 4.1: Partition by Cylinder Sets
• proof