583 problems

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

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP}^2$ be a dominant rational self-map defined over $\mathbb{Q}$, and let $P=[x_0:y_0:z_0]$ be a generic rational point. Write $f^n(P)$ f…

Eremenko's conjecture. Each connected component of $I(f)$ is unbounded.

In the space of rational maps of a fixed degree, a hyperbolic rational map is a rational map for which every critical orbit converges to an attracting cycle. Fatou's density conjec…

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…

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

Let $f_c(z):=z^2+c$ for $c\in\mathbb{C}$, and let $\mathcal{M}$ be the set of parameters $c\in\mathbb{C}$ for which the orbit of the critical point $z=0$ does not escape to infinit…

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…

For each integer $n\geq 1$, let $\mathrm{Per}_n(0)\subset \mathrm{M}_2$ be the PCF-special curve parametrizing quadratic rational maps for which one critical point has exact period…

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 $M_1=M(\{0,\pm1\})$ be the closure in $\mathbf{C}$ of the set of zeros of polynomials with coefficients in $\{0,\pm1\}$ and nonzero constant term. Write…

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…

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…

Bedford's conjecture. The basin $\Omega_{\{F_j\}}$ is biholomorphic to $\mathbb{C}^k$. While the conjecture remains open, it is known that $\Omega_{\{F_j\}}$ is an increasing union…

Let $\mathcal{F}_a$ be the correspondence in the family described above, $\mathcal{M}$ the Mandelbrot set, and $\mathcal{M}_{\Gamma}$ the modular Mandelbrot set, namely the set of…

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 $f$ be a rational map, let $J(f)$ denote its Julia set, and let $\widehat{J}(f)$ be the closure of the grand orbit of the singular values and periodic points. The Teichmüller f…

A rational map is a holomorphic map $f \colon \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$ of degree at least $2$; its multipliers are the derivatives associated with its…

A transcendental entire function is a transcendental holomorphic map $f:\mathbb{C}\to\mathbb{C}$; its order is the usual growth order of an entire function, and a Fatou component i…

Let $\mathcal{F}$ be a generic holomorphic foliation on the complex projective plane $\mathbb{CP}^2$. A leaf of $\mathcal{F}$ is a Riemann surface, and a topological cylinder is a…

Let $f$ be a rational map of degree at least two. A point of the Julia set $J(f)$ is buried if it does not lie on the boundary of any component of the Fatou set. Makienko's conject…

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…

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

Attracting-orbit conjecture. The only attracting periodic points for $h_{19}$ are the elements of ${\mathcal{O}_{72}}$. Moreover, the union of the basins of attraction for…

Let $\mathcal{C}_d$ be the connectedness locus of the space $\mathcal{P}_d$ of degree-$d$ polynomials, and let a polynomial in $\mathcal{C}_d$ be combinatorially rigid if it has no…