Non-existence of wandering intervals for asymmetric unimodal maps

TL;DR

The paper proves that asymmetric unimodal maps with nonequal critical orders $\ell_- \neq \ell_+$ on $[0,1]$ admit no wandering intervals. It combines cross-ratio distortion estimates and the Koebe principle with a careful analysis of closest-returns to the critical value, introducing and controlling quantities $\eta_k$ that compare the left/right endpoint distances to the critical orbit. By establishing a cascade of lemmas (A–C) that force $\eta_k$ to decay too rapidly relative to endpoint gaps, the authors derive a contradiction to the assumed wandering interval, hence a contraction principle for preimages of small intervals and robustness of the system’s non-wandering structure. The result extends the Denjoy-type absence of wandering intervals to a broad class of asymmetric unimodal maps and enriches the understanding of one-dimensional dynamics near critical points.

Abstract

We prove that an asymmetric unimodal map has no wandering intervals.

Figures (3)

• Figure 1: Graphic of an asymmetric unimodal map
• Figure 2: On top, the double lined interval denote $B_k$. On the bottom, we have the interval $B_k^{s(k-1)}$. Dashed lines represent closest return of $Q$ to $c_1$, and pre-closest return around $c$.
• Figure 3: On top, the double lined interval denote $f^{s(k)-s(k-1)-1}(H_k)$. On the bottom, we have the interval $H_k$.

Theorems & Definitions (14)

• Corollary 1.1
• Definition 2.1
• proof : Completion of proof of the Main Theorem
• Lemma 3.1
• proof
• Theorem 1
• Theorem 2: One-sided Koebe principle
• Lemma 3.2
• proof
• proof : Proof of Lemma A