Thurston algorithm for power-law lift families

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.

Let $I=[0,1]$ and fix the number $n$ of laps (maximal monotonicity intervals) and a boundary map $\mu:\{0,1\}\to\{0,1\}$. Write $\mathcal P(n,\mu)$ for the continuous piecewise monotone interval maps with those data. Starting with $f_0$, lift $f_k$ to the degree-$n$ polynomial $p_k\in\mathcal P(n,\mu)$ whose critical points lie in $I$ and whose ordered critical values equal those of $f_k$. Let $h_k:I\to I$ be the homeomorphism taking ordered critical points of $f_k$ to those of $p_k$ with $p_k\circ h_k=f_k$, and put \[f_{k+1}=h_k\circ p_k=h_k\circ f_k\circ h_k^{-1}.\] This is the real Thurston algorithm in the source. A kneading sequence records itineraries of critical values relative to the ordered laps and their boundaries. A map is postcritically finite when its critical points have finite forward orbits. For a different lifting family, replace the polynomial lifts by members of that family having the same ordered critical data and the analogous homeomorphisms.For the unimodal family in this question take \[p_{k,\alpha}(x)=k-k|2x-1|^\alpha,\qquad \alpha>1,\quad 0<k\leq1.\] Both endpoints map to $0$. The subscript $k$ in this family is a parameter and is separate from the iteration index in the algorithm.

Statement recorded in the supplied source.

For each $\alpha>1$, does the real Thurston algorithm with lifts in this family converge whenever the initial map has periodic or preperiodic kneading sequence? [S2]

Short English statement

Does the algorithm converge for every power-law lift family with exponent greater than one and every periodic or preperiodic kneading sequence?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300076 (AMR-052-0076). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld (editor) and conference contributors, Conformal Dynamics Problem List, arXiv:math/9201271v1 (1990), §3, Question 2 and preceding family definition, printed p. 9. https://arxiv.org/abs/math/9201271v1.