Convergence of the real Thurston algorithm

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.

Statement recorded in the supplied source.

Formulate and prove precise hypotheses under which the sequences $p_k$ and $f_k$ converge to a common map with the kneading data of $f_0$, including settings in which $f_0$ is not postcritically finite. [S2]

Short English statement

Find precise convergence conditions for the real Thurston algorithm, including beyond finite critical orbits.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300075 (AMR-052-0075). 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 1 and preceding construction, printed pp. 8--9. https://arxiv.org/abs/math/9201271v1.