18 problems

Let $T$ be a trellis and let $f$ be a trellis map for $T$. Write $N_\infty(f)$ for the asymptotic Nielsen number and $h_\mathit{top}$ for topological entropy. Optimality conjecture…

Let $M$ be a non-orientable closed surface, and let $r\ge 2$. Write $\mathfrak{X}^r(M)$ for the space of $C^r$ vector fields on $M$. A vector field is Morse–Smale if its singularit…

Let $h$ be a homeomorphism with module $d$, and let $W_{0 \rightarrow \infty}$ and $W_{\infty \rightarrow 0}$ denote the corresponding dynamical sets. A heteroclinic branch is an…

General Floer homology and asymptotic growth conjecture.

Bounded convex decomposition conjecture. There exists a map

Bounded-deviation strengthening. There exists $C>0$ such that if an orbit $\widetilde y,\dots,\widetilde f^{n_0}(\widetilde y)$ of $\widetilde f$ crosses $V_C(\widetilde\gamma)$ fo…

Let $f$ be a surface symplectomorphism, and interpret “typical” in the conjectural generic or parameterwise sense under discussion. Positive metric entropy conjecture. A typical su…

Exact minimal stretch factor conjecture. The normalized stretch factor of $f$ is greater than or equal to the largest real root of

McMullen's conjecture. Every pseudo-Anosov automorphism $f\colon S\to S$ admits a lift $f'\colon S'\to S'$ to some finite cover $S'\to S$ such that…

Let $\phi$ be a pA mutation loop on a marked surface $\Sigma$. Write $\mathcal{E}_\phi^a$ for its associated asymptotic entropy and $\lambda_{\pi(\phi)}$ for the spectral radius of…

Tresser's conjecture. Generically, maps in the boundary of positive topological entropy have a set of periodic orbits which, except for a finite subset, consists of infinitely many…

Let $S_g$ be a closed orientable surface, let $f\in\operatorname{Mod}(S_g)$ be pseudo-Anosov with stretch factor $\lambda$, and let $C$ be the constant appearing in the small-stret…

A bi-Perron number is a unit algebraic integer $\alpha>1$ such that every Galois conjugate other than $\alpha$ lies in the annulus … except possibly for $\alpha^{-1}$. A stretch fa…

Let $\Gamma\subset \operatorname{SL}(3,\mathbb {R})$ be a lattice, let $S$ be a closed connected surface, and let $\Gamma$ act on $S$ by $C^{1+\beta}$ diffeomorphisms. Assume that…

Symplectic Conley conjecture. Every such symplectomorphism has infinitely many simple periodic orbits.

Distortion conjecture. An element in $\mathrm{Homeo}_{0}(S)$ which satisfies the conclusion of the proposition—that is, is non-spreading—is distorted.

Conjugacy-class conjecture. The closure of the conjugacy class of any pseudo-rotation of the torus of angle $\alpha$ contains the translation of angle $\alpha$. The closure of the…

Let $A$ be an expansive attractor of a homeomorphism of a compact surface, where an attractor is understood in the setting above and $A$ has the local product structure established…