17 problems

Let $\alpha$ be a $\mathbb{Z}_+^2$-action on the circle $\mathbb{S}^1$ whose generators are … where $p$ and $q$ are coprime integers. Let $\sigma_\alpha$ be the shift map on the in…

Let $\mu$ be the unique normalized invariant measure of a cubic critical homeomorphism of the circle, and let $HD(\mu)$ denote the Hausdorff dimension of its measure-theoretical su…

A hyperbolic-like action of a group on the circle is an action by orientation-preserving circle homeomorphisms in which every non-trivial element has exactly two fixed points, one…

Let $G\leq \operatorname{Homeo}_+(S^1)$ be a subgroup such that every non-trivial element has at most two fixed points, and whose action on $S^1$ is minimal. Assume that the action…

Let $f\in {PL}_+S^1$ be a piecewise linear orientation-preserving circle homeomorphism that maps $\mathbb{Q}$ to $\mathbb{Q}$ and whose slopes are powers of a single rational numbe…

In a typical monotone one-parameter family of multicritical circle maps, the parameters with irrational rotation number form a Cantor set, called the mode-locking Cantor set. Unive…

A real-analytic circle diffeomorphism may have a Diophantine rotation number without being assumed to be a small perturbation of a rigid rotation. Arnold's conjecture. No restricti…

Let a higher rank lattice be a lattice in a simple Lie group with finite center and real rank at least $2$. Consider a continuous action of such a lattice on the circle…

Geometricity conjecture. If $\rho$ is fully rigid, then $\rho$ is geometric.

Let $G\leq \operatorname{Diff}_+^2(\mathbf{S}^1)$ be a finitely generated group whose action on the circle has an exceptional minimal set $\Lambda$. Ghys–Sullivan–Hector conjecture…

Let $G\leq \operatorname{Diff}_+^2(\mathbf{S}^1)$ be a finitely generated group whose action on the circle is minimal. Ghys–Sullivan conjecture. The action is Lebesgue ergodic: eve…

Let $G\leq \operatorname{Diff}_+^\omega(\mathbf{S}^1)$ be a finitely generated one-ended group that is not finitely presented or has elements of unbounded torsion. Expanding-action…

Let $G\leq \operatorname{Diff}_+^1(\mathbf{S}^1)$ act minimally on the circle, and let $\mathrm{NE}$ be its set of non-expandable points. Property $(\star)$ means that for every…

Let $G\leq \operatorname{Diff}_+^\omega(\mathbf{S}^1)$ be a finitely generated locally discrete group. A locally discrete group's structure conjecture predicts both that the action…

Period-set characterization conjecture. There exist sets $E_c,E_d\subset\mathbb N$, each a finite union of tails of the orderings $\leso{2}$ and $\leso{3}$, such that

For any real number $\theta$, write $\{\theta\}$ for its fractional part. Let $d=1$, let $w_1$ be any real frequency, and form the ordered fractional parts arising from the relevan…

Let $\Gamma$ be a higher-rank lattice acting continuously on the circle $S^1$. The circle-action conjecture. Every such action should be finite. This is part of the low-dimensional…