15 problems

Let $f$ be a smooth diffeomorphism of the circle with a Diophantine rotation number, meaning that there exist $C>0$ and $\tau>0$ such that for every $k\in\mathbb{Z}$,…

Let $\alpha\colon\mathbb Z^k\to\operatorname{Diff}^{\infty}_+(S^1)$ be generated by commuting diffeomorphisms $f_1,\dots,f_k$, with at least one irrational rotation number…

Let $M$ be a phase space, and let $F=(f_\beta)_{\beta\in B}$ and $G=(g_{\tilde{\beta}})_{\tilde{\beta}\in\tilde{B}}$ be families of diffeomorphisms of $M$. They are weakly topologi…

Let $f\colon\mathbb R/\mathbb Z\to\mathbb R/\mathbb Z$ be an analytic orientation-preserving circle diffeomorphism, let $F$ be a lift of $f$ to the real line, and let…

Let $f$ be a sufficiently regular circle diffeomorphism with irrational rotation number $\rho$, and let $x\mapsto x+\rho$ be the rigid rotation with the same rotation number. Arnol…

Let $G\subset \operatorname{Diff}_+^\omega(\mathbb S^1)$ be a locally discrete, finitely generated subgroup. The dichotomy in the classification theorem stated immediately before t…

Renormalisation hyperbolicity conjecture. The renormalisation operator $\mathcal{R}$, restricted to $\mathcal{E}_{\mathbf{c}}$, is hyperbolic when restricted to its attractor.

Rigidity conjecture. There is a full-measure set of decorated rotation numbers $\gamma$ such that, whenever $T_1$ and $T_2$ are circle diffeomorphisms with breaks having breaks of…

Let $G$ be a locally discrete subgroup of $\mathsf{Diff}_+^\omega(\mathbb{S}^1)$, where $\mathsf{Diff}_+^\omega(\mathbb{S}^1)$ denotes the group of orientation-preserving real-anal…

Naive conjecture. Positive asymptotic distortion should imply that $f$ has hyperbolic periodic points.

Let $F$ be a lift of a circle diffeomorphism and let $\tau(F+\omega)$ denote its complex rotation number for $\omega\in\mathbb H$. For a real parameter $\omega_0$, suppose that the…

Dippolito's conjecture. The restriction of the action of $G$ to $\Lambda$ should be topologically conjugate to the action of a group of piecewise-affine homeomorphisms.

Let a finitely generated group of $C^2$ circle diffeomorphisms admit an exceptional minimal set $\Lambda$, meaning a minimal closed invariant subset of the circle that is homeomorp…

Let a finitely generated group act smoothly and minimally on the circle, meaning that every orbit is dense. Let Lebesgue measure denote the standard Lebesgue measure on the circle.…

Let $\Gamma$ be a finitely generated group acting minimally on the circle $\mathrm{S}^1$ by $C^r$ circle diffeomorphisms, with $r\geq 2$. Ergodicity is with respect to Lebesgue mea…