Homeomorphism type of exponential Knaster continua

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a real parameter $\lambda>1/e$, let $E_\lambda(z)=\lambda e^z$ on $\mathbb C$, and let $E_\lambda^n$ denote the $n$th iterate, with $E_\lambda^0$ the identity. Define $C_\lambda=\{z\in\mathbb C:0\leq\operatorname{Im}(E_\lambda^n(z))\leq\pi\text{ for every }n\geq0\}$. Give $C_\lambda$ its subspace topology from the plane. A homeomorphism is a continuous bijection with continuous inverse. The source describes these explicitly defined sets as Knaster-like continua.

Statement recorded in the supplied source.

For every pair of real parameters $\lambda,\mu>1/e$, are $C_\lambda$ and $C_\mu$ homeomorphic? [S2]

Short English statement

Do the explicitly defined invariant strip sets have the same topology for all real exponential parameters greater than $1/e$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300063 (AMR-052-0063). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Robert Devaney, "Open Questions in Non-Rational Complex Dynamics", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), third problem in Section 3 (Knaster-like continua), pp. 36; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.