15 problems

Let $f\colon \widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be a post-critically finite rational map that is not a flexible Lattès map. A simple closed curve on…

Let $f:S^1\to\mathbb{CP}^1$ be a CaTherine wheel arising from an expanding Thurston map. Uniform quasiconformality conjecture. There exists a finite constant $K$ such that $f$ is a…

Let $B$ be a non-exceptional Hurwitz biset, and let $B^*$ denote its associated semigroup of elements. An element of $B^*$ is unobstructed if it has no Thurston obstruction. Conver…

Let $f\colon (\widehat{\mathbb{C}},P)\to(\widehat{\mathbb{C}},P)$ be a rational Thurston map with a hyperbolic orbifold. Let $\overline{\mathscr{C}_P}$ be the set of isotopy classe…

Let $f \colon (S^2,A) \righttoleftarrow$ be a Thurston map with marked set $A$, and let $m\geq 1$. A Thurston map is realized when it is equivalent to a rational map with the presc…

Let $(F,Q)$ be a marked expanding Thurston map, where $F$ is an expanding Thurston map of the sphere and $Q$ is the marked set. An $F$-invariant graph is a graph preserved by $F$.…

Higher-dimensional extension conjecture. Theorem on X-rays holds with respect to any fixed point $\star$ of $F$.

Extension conjecture. Theorem on X-rays and the attractor corollary should hold for all post-critically finite non-Lattès maps.

An expanding Thurston map is a branched covering of the sphere whose dynamics admit a suitable expanding metric, and an anti-mating is a function associated with a pullback curve w…

Let $f\colon S^2\to S^2$ be a Thurston map with a hyperbolic orbifold that is realized by a rational map. Let $P_f$ denote its postcritical set, and let pullbacks be considered up…

Let $f$ be a rational Thurston map, and consider the pullback relation on nontrivial curves associated to $f$. A finite global attractor conjecture asserts that, if $f$ is not a fl…

For a Thurston map $g$ with canonical obstruction $\Gamma\neq\emptyset$, let $g_\Gamma$ be the associated decomposition, and consider its first-return maps that are Thurston maps.…

Exceptional Hurwitz structure set conjecture. The Hurwitz structure set of every primitive NET map with constant pullback map is either an MC2 or MC4 Hurwitz structure set, or it i…

Extended half-space algorithm conjecture. The extended half-space algorithm decides, in finite time, whether or not $f$ is equivalent to a rational function.

Cellular Markov partition conjecture. Every expanding Thurston map admits a cellular Markov partition.