18 problems

Non-degeneracy conjecture. All frequency-locking intervals in the family away from $\beta=0$ are non-degenerate. Equivalently, at the boundaries of each phase-locking interval, the…

Let $f:S^1\to S^1$ be an analytic circle map, and let $\tau(f)$ denote its rotation number. Arnold conjectured that if $\tau(f)$ belongs to a set of full Lebesgue measure, then the…

Universal Hölder rigidity conjecture. In the general case, the conjugacy is always $C^{1+\alpha}$ at the critical point, for some universal $\alpha$ independent of the rotation num…

Universality conjecture. When $\nu>2$, the parameter scalings $\delta_n$ tend to a universal limit depending only on $\nu$.

Let $F:\mathbb{S}^1\to\mathbb{S}^1$ be a critical circle mapping with rotation number $\theta$. Let $H:\mathbb{S}^1\to\mathbb{S}^1$ be the normalized linearizing map satisfying … a…

Let $f$ be a real analytic critical circle map with irrational rotation number $\rho(f)=[a_0,a_1,a_2,\dots]$, and let $h:\mathbb{S}^1\to\mathbb{S}^1$ be the normalized linearizing…

Brezis's conjecture. There exist constants $C>0$ and $\delta_0>0$ such that the displayed inequality holds for every continuous $f\in C^0({\mathbb S}^1;{\mathbb S}^1)$ and every…

Fix $gamma>1$ and an irrational rotation number $rho$ with $0<rho<1$. A smooth $gamma$-critical circle map is a smooth critical circle map whose critical point has order $gamma$. N…

A critical circle map is a circle map with a critical point whose order is characterized by $gamma>1$. Two such maps have the same criticality type when they have the same order…

Let $f$ and $g$ be $C^3$ irrational multicritical circle maps with the same number of critical points. Their signature consists of the data determining the critical-point configura…

Let the up flow have a generic tangency with the upper threshold, producing a gap, and let the down map map both endpoints of this gap into the pre-image under the up map associate…

Let $R(t)$ be the access map associated with the delay system, and let $R^q(t)$ denote its $q$-fold iterate. The Lyapunov spectrum is considered in the limit of large Lyapunov indi…

Rigidity conjecture. If $f$ and $g$ have the same signature, then they are conjugate by a $C^1$ diffeomorphism. Moreover, if their common rotation number is of bounded type, then t…

A critical circle map is a circle map with at least one point where its derivative vanishes. Finite cyclicity conjecture. A generic family of critical circle maps should have finit…

Geometric rigidity conjecture. Any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other…

Let the boundary of the set of hyperbolic systems be considered in a natural parameter or measure-theoretic setting. A Kupka–Smale diffeomorphism is a diffeomorphism whose periodic…

For $(a,b) in (0,1)times mathbb{R}$, let $f{a,b}(x)=x+a+bsin(2pi x)$, and let its rotation interval be the interval of rotation numbers associated with the map. Fix an interval who…

Let $\Omega_{F-B}$ be the subfractal of the $F-B$-structured interval associated with the Farey–Brocot measure, characterized by $f(\alpha)=\alpha$ and $f'(\alpha)=1$. Let … Its en…