247 problems

Let $X$ be a compact connected complex manifold of complex dimension $n$, and let $f:X\to X$ be a holomorphic Anosov diffeomorphism. Thus there is an invariant splitting…

For every $c\in\mathbb{C}$, let $f_c(z)=z^2+c$ and let $P(f_c)=\overline{\{f_c^n(0):n\ge 1\}}$ be its postcritical set. The conjecture asserts that $m_2\bigl(P(f_c)\bigr)=0$, where…

Does there exist a transcendental entire function $f:\mathbb{C}\to\mathbb{C}$ with a wandering Fatou component $U$ such that the forward orbit of $U$ is bounded, i.e.…

Let $p\in\mathbb{C}[z]$ be a centered polynomial of degree at least $2$, let $Z(p)$ denote its set of roots, and let $C_p$ be the Chebyshev map…

Almost-everywhere zero-measure conjecture. For Lebesgue almost all $u$, $\mathcal J_u$ has zero Lebesgue measure.

Let $X$ be a complex manifold endowed with a Riemannian metric, and let $f\colon X\to X$ be an automorphism acting hyperbolically on an invariant compact subset $K\subset X$. For…

Let $0\le s\le k$ be an integer. Let $f:\mathbb{P}^k\to\mathbb{P}^k$ be a holomorphic endomorphism of degree $d\geq 2$, and let $T$ be its Green $(1,1)$-current. For an analytic su…

Let $f_{a,b}(z)=z^{3}-3a^{2}z+2a^{3}+b$ be a monic, reduced cubic polynomial with marked critical point $a$, and let $\a7\Sigma_{k,n}$ be the Zariski closure in the moduli space…

Let $f\colon \widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be a post-critically finite rational map that is not a flexible Lattès map. A simple closed curve on…

The Mandelbrot set $\mathcal{M}$ is the connectedness locus of the quadratic family $p_c(z)=z^2+c$. MLC conjecture. The Mandelbrot set $\mathcal{M}$ is locally connected. Local con…

For a quadratic polynomial, let its core entropy be the entropy of the dynamics on its Hubbard tree, and let the polynomial vary in the space of quadratic polynomials equipped with…

Repelling 45-lines conjecture. On the $45$-lines $h_{19}$ is repelling and, hence, $\{X=0\}$ resides in the Julia set $J_{h_{19}}$.

Positive-measure disconnectedness conjecture. There exists a set of positive Lebesgue measure $\Lambda\subset\mathbb T$ such that $\mathcal J_u$ is disconnected for all…

Let $p$ be a normalized polynomial, let $n\geq 2$, and let $K_{p,n}$ be the König root-finding map. Write $\Sigma p$ for the symmetry group associated with $p$ and $\Sigma K_{p,n}$…

Density of hyperbolicity conjecture. The set $\mathcal{H}$ is dense in $\mathcal{M}$.

Let $f$ be a transcendental entire function, and let … be its escaping set. Eremenko's conjecture. Every connected component of $I(f)$ is unbounded. The conjecture has been confirm…

Positive Lyapunov exponent conjecture. One has $\Lambda>0$.

Let $P$ be a postcritically-finite non-renormalizable quadratic polynomial whose Julia set $\mathcal{J}_P$ is a dendrite. Intermediate-growth conjecture. The iterated monodromy gro…

Let $\mathcal{M}$ be the Mandelbrot set, and write … where $J_c$ is the Julia set of the quadratic map with parameter $c$. Star-shapedness conjecture. The set…

Let ${\mathcal{M}^p}$ be the generalized Mandelbrot set for the polynomial $Q_{p,c}(z)=z^p+c$, where $z,c\in\mathbb{C}$ and $p\geq 2$ is an integer. The real-slice conjecture. If…

Jordan-domain conjecture. Every bounded immediate attracting or parabolic basin of a transcendental entire function is a Jordan domain.

Let $R$ be a rational map of the Riemann sphere, and let its Julia set be the set on which the associated dynamics is chaotic. An invariant line field for $R$ is a Beltrami form…

Makienko's conjecture. The Julia set $J(R)$ has buried points if and only if there is no completely invariant component of the Fatou set of $R^2$.

Let ${\cal P}_0$ be the collection of chess squares in the level-zero puzzle for the Feigenbaum map, and let $\operatorname{diam}(P)$ denote the diameter of a chess square $P$. Uni…

Let $Y$ be a smooth projective variety and $f:Y\dashrightarrow Y$ be a rational dominating map. A smooth birational model is a smooth projective variety $\hat Y$ together with a bi…