Mathematics Problem Archive

For $f_\lambda(z)=\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove tha...

For $f_\lambda(z)=\lambda z^2+z^3$, let $U$ be the parameter component where both finite critical points lie in the immediate basin of zero. Prove tha...

Let $f_i(z)=z^2+c_i$ range over finitely many critically periodic quadratic polynomials, let $g_n$ be the iterated tuning $f_1\vdash\cdots\vdash f_n$,...

For $P_\rho'(z)=\lambda(1-z)^\rho$, $P_\rho(0)=0$, let $h$ linearize the Siegel disk and write $h'(\zeta)/(1-h(\zeta))=\sum_{\nu\ge0}a_\nu\zeta^\nu$. ...

For $P_\alpha(z)=z^2+e^{2\pi i\alpha}z$ with a Cremer fixed point at $0$, is there an arc in its Julia set joining $0$ to its preimage $-e^{2\pi i\alp...

Give a plausible topological model for the Julia set of a Cremer polynomial....

Produce a reliable computer picture of the Julia set of a Cremer polynomial....

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?...

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 $U$?...

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?...

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

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

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-...

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

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....

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...

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