45 problems

Marmi–Moussa–Yoccoz conjecture. The function $\Upsilon$ is $(1/2)$-Hölder continuous. This conjecture concerns the regularity of the correction to the Bryuno asymptotic for the con…

Let $M$ be a smooth manifold, let $f\in\operatorname{Diff}^\infty(M)$ be an Anosov diffeomorphism, and let $\mathcal{J}_M(f)$ denote the set of complex structures on $M$ preserved…

Let $(\phi_t)$ be a parabolic semigroup of zero hyperbolic step in the unit disk, with Denjoy–Wolff point $\tau$. The slope is the cluster set of $\arg(1-\overline{\tau}\phi_t(z))$…

Let $f:\mathbb {P}^k\to\mathbb {P}^k$ be a holomorphic endomorphism of degree $d\ge 2$, and let $\mu$ be its equilibrium measure. Let … be the Lyapunov exponents of $\mu$. Binder–D…

Symmetry conjecture. Non-injectivity only happens when $f$ or a power of $f$ has a Möbius symmetry.

Rigidity conjecture. The conjectured rigidity assertion is not recoverable from the supplied statement span.

Let $\phi$ be an analytic self-map of the unit disk $\mathbb{D}$, and write $\phi_n=\phi^{\circ n}$ for its $n$-th iterate. Suppose that $\phi$ is type II parabolic, meaning that i…

Let $\delta$ be an irrationally indifferent point, and let $\mathcal{P}$ denote the set of properties involved in the local linearization around $\delta$. Finiteness conjecture. Th…

Let $\mathscr{X}$ be a taut, acyclic complex manifold, meaning that $H_j(\mathscr{X},\mathbb{Z})=0$ for all $j\in\mathbb{N}$, and let…

Fatou's conjecture. Functions in $\mathcal{S}$ admit no invariant line fields supported on their Julia sets.

Bullett–Penrose conjecture. The connectedness locus of this family of algebraic correspondences is homeomorphic to the Mandelbrot set.

Let $\Omega\subset \mathbb C^N$ be a bounded convex domain, with $N\geq 2$. A bounded convex domain is visible when it satisfies the visibility property for complex geodesics, and…

Unbounded-boundary-orbit conjecture. Every unbounded orbit on the boundary of a heteroclinic region, and in particular the red orbits $C_3,\ldots,C_7$, is a separatrix.

Existence conjecture for polynomial flows with prescribed heteroclinic regions. For all $n,m\in\mathbb{N}$, there exists a complex polynomial $F_{n,m}$ of degree $n+m$ such that th…

Let $M$ be a Kähler manifold and let $f$ be a holomorphic partially hyperbolic diffeomorphism of $M$, with center distribution $E^c$. Kähler center-holomorphicity conjecture. The c…

Let $f$ be a holomorphic partially hyperbolic diffeomorphism, with center distribution $E^c$. Real-analyticity conjecture. The center distribution $E^c$ is real analytic. The conje…

Let $f$ be a holomorphic partially hyperbolic diffeomorphism on a complex $3$-manifold. Three-dimensional classification conjecture. Up to passing to a finite cover, $f$ is holomor…

Let $X$ be a bounded domain in $\mathbb{C}^{N}$ equipped with the Kobayashi metric, and assume that it is a proper metric space. Let $\overline{X}\subset\mathbb{C}^{N}$ be the stan…

Let $(\phi_t)$ be a non-elliptic semigroup in $\mathbb{D}$ with Koenigs function $h$, and let $\gamma:[0,+\infty)\to\mathbb{D}$ be a regular backward orbit converging to a super-re…

Homogeneous-chambar singular-set conjecture. For every $k\geq 1$,

4-chambar classification conjecture. Up to affine conjugacy, every $4$-chambar on an open subset of $\mathbb{C}$ is one of the following types:

Semi-simple vector-field conjecture. Any such $X$ is not almost a $p$-chambar.

Let $\mathbb{D}$ be the open unit disk, and let $\varphi$ be an essentially linear fractional self-map of $\mathbb{D}$ with exactly one fixed point $w$ in $\overline{\mathbb{D}}$.…

Let $f$ be a post-critically algebraic endomorphism of ${\mathbb{CP}}^n$ with $n\geq 2$ and degree $d\geq 2$. Let $\lambda$ be an eigenvalue of $f$ along a periodic cycle. Hyperbol…

Invariant-component bound. It is conjectured that the number of completely invariant components is at most one for transcendental entire functions and at most two for meromorphic f…