Mathematics Problem Archive

If a quadratic polynomial $f_c$ is infinitely renormalizable of bounded type, must $J(f_c)$ be locally connected? In particular, is the Julia set of t...

For every real $c\in[-2,1/4]$, is the Julia set of $f_c(z)=z^2+c$ locally connected?...

Does every nested intersection $\bigcap_k H_1*\cdots*H_k*M$ of tuned copies of the Mandelbrot set consist of one point? Equivalently, are infinitely r...

For the Mandelbrot limb $M(p/q)$ of internal angle $p/q$, is $\operatorname{diam}M(p/q)<K/q^2$ for an absolute constant $K$? If not, is it at least bo...

Can a nowhere-dense Julia set of a rational map have positive Lebesgue measure?...

If a rational map $f$ has $J(f)=\widehat{\mathbb C}$, is $\omega(z)=\widehat{\mathbb C}$ for almost every $z$, and is $f$ conservative with respect to...

If $J(f)=\widehat{\mathbb C}$, is $f$ ergodic for Lebesgue measure? At least, does it have at most $2\deg f-2$ ergodic components?...

Are Lattès maps the only rational maps having measurable invariant line fields on their Julia sets?...

Find an explicit rational map whose Julia set has Hausdorff dimension two. When such a Julia set has zero Lebesgue measure, identify a natural geometr...

For an analytic family $\mathcal A$ of rational maps, let $Q\subset\mathcal A$ be the $J$-unstable locus. What is the Lebesgue measure of $Q$, and is ...

For a geometric coding tree of inverse branches of a holomorphic map, let $z_\infty:D(z_\infty)\to\overline U$ map each convergent symbolic branch to ...

Let $f:U\to f(U)$ be a proper holomorphic map of degree at least two on a simply connected attracting basin $U$, and suppose $f$ extends holomorphical...

In the setting of Przytycki Problem 1.1, is every $x\in\partial U$ with $\liminf_{n\to\infty}n^{-1}\log|(f^n)'(x)|>0$ accessible from $U$?...

In the setting of Przytycki Problem 1.1, is $h_{\mathrm{top}}(f|_{\partial U})=\log\deg(f|_U)$?...

Can the boundary of a Siegel disk contain periodic points or points with positive Lyapunov exponent? Must the topological entropy of the boundary dyna...

For a holomorphic quasi-repeller $\Lambda$, is every invariant ergodic measure on $\overline\Lambda$ the image of a measure on a one-sided shift under...

Characterize the positive-entropy invariant measures $m$ on a holomorphic quasi-repeller for which the almost-sure invariance principle, law of the it...

For a positive-entropy invariant measure $m$ on a holomorphic quasi-repeller $\Lambda$, is $m$ absolutely continuous with respect to Hausdorff measure...

For which positive-entropy invariant measures $m$ does failure of uniform $L^2(m)$ boundedness of the sums of $\log\operatorname{Jac}_m f-\kappa\log|f...

For which positive-entropy invariant measures $m$ does uniform $L^2(m)$ boundedness of the sums of $\log\operatorname{Jac}_m f-\kappa\log|f'|$, where ...

Which theorems about boundary behavior of Riemann maps have analogues for geometric coding trees?...

For a holomorphic quasi-repeller $\Lambda$, is $\sup_{m\in\mathcal M^+(\Lambda)}\dim_Hm=\dim_H\overline\Lambda$? Does allowing all invariant ergodic m...

Find a collection of representative examples of transcendental entire maps whose dynamics may serve as models for general phenomena....

Describe the dynamics of the entire maps $z\mapsto\lambda e^z\sin z$ and $z\mapsto\lambda e^z\cos z$....

For $E_\lambda(z)=\lambda e^z$, characterize completely the parameters $\lambda$ for which $J(E_\lambda)=\mathbb C$....

Many parameters with $J(E_\lambda)=\mathbb C$ lie on parameter curves or hairs. Are these hairs $C^\infty$? Are they analytic?...

For parameters $\lambda,\mu>1/e$, are the Knaster-like continua arising in the dynamics of $E_\lambda(z)=\lambda e^z$ and $E_\mu(z)=\mu e^z$ homeomorp...

Describe the parameter-space structure for the entire families $C_\lambda(z)=\lambda\cos z$ and $S_\lambda(z)=\lambda\sin z$....

Determine the measure and Hausdorff dimension of the parameter hairs in the exponential, sine, and cosine families....

Describe the dynamics of Newton's method when applied to broad natural classes of transcendental entire functions....

Does there exist an entire function with a wandering Fatou component whose orbit of components is bounded?...

Let $\varphi$ be a holomorphic germ fixing $z_0$ with multiplier $e^{2\pi i\alpha}$ for irrational $\alpha$. Can $\varphi^n(z)\to z_0$ uniformly on so...

Under the hypotheses of Eremenko–Lyubich Question 2, can even a single orbit converge to $z_0$?...

Call a polynomial bad if its Newton map has an attracting cycle that is not a root. Prove that every bad degree-$d$ polynomial $f_1$ belongs to a one-...

Let all roots of a degree-$d$ polynomial $f$ lie in the unit disk, let $\alpha$ be a root of multiplicity $m$, and let $A^*_{h}(\alpha)$ be its immedi...

For $P_c(z)=z^2+c$ with connected Julia set, characterize the parameters $c$ for which $J(P_c)$ is locally connected....

If the quadratic polynomial $P_c(z)=z^2+c$ is nonrecurrent, does there exist a conformal metric $\rho(z)|dz|$ with integrable singularities in which $...

For monic polynomials $z^n+a_{n-1}z^{n-1}+\cdots+a_1z$ with $|a_1|\ge1$, external rays landing at the fixed point $0$ have a rotation number. Does thi...

For a piecewise monotone interval map, iteratively replace its critical values by those of a polynomial with the same ordered critical data and conjug...

For the lift family $x\mapsto k-k|2x-1|^\alpha$, $\alpha>1$, does the real Thurston algorithm converge whenever the initial interval map has a periodi...

Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for every periodic or preperiodic kneading sequ...

Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for arbitrary kneading sequences....

Let $f$ be a polynomial diffeomorphism of $\mathbb C^2$ with Jacobian determinant $\delta$, let $U$ be a component of the interior of the bounded-forw...

In the rank-zero case, if the limiting map on an invariant stable component is constant with value $x_0\in\partial U$, prove that one eigenvalue at $x...

Can the subsequential limit map on an invariant stable component of a polynomial diffeomorphism of $\mathbb C^2$ be a retraction onto a Herman ring or...

In the rank-two case for a polynomial diffeomorphism of $\mathbb C^2$, can an invariant stable component be a product of two Herman rings, or a produc...

For every degree $d$, prove that expanding (hyperbolic, Axiom A) maps are dense in the spaces $\operatorname{Rat}_d$ of rational maps and $\operatorna...

For a geometrically finite rational map $f$, prove that either its Julia set is the whole sphere and $f$ is ergodic there, or its Julia set has Hausdo...

Prove that every connected component of the Julia set of a geometrically finite rational map is locally connected....

Develop an analogue of the Haken decomposition for geometrically finite rational maps. In particular, if the Julia set is disconnected, can the map be...