Mathematics Problem Archive

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...

Extend Thurston's finite combinatorial classification from critically finite rational maps to all geometrically finite rational maps: give finite topo...

If a complete hyperbolic $3$-manifold $N$ has fundamental group generated by $n$ elements, is there a bound $R_n$, depending only on $n$, on the radiu...

For $n>1$, do there exist nontrivial critically finite rational maps $f:\mathbb P^n\to\mathbb P^n$ whose postcritical hypersurface $V$ has Kobayashi-h...

Let $A$ be a hyperbolic attractor of a diffeomorphism of a compact $3$-manifold. Beyond the known Anosov, laminated, Williams, and invariant-torus cas...

Given an explicitly specified smooth orientation-preserving diffeomorphism $F$ of the $2$-sphere, is its topological entropy Turing-computable to arbi...

Show that a random walk on the mapping class group gives a pseudo-Anosov element whose invariant foliations have generic trivalent singularities with ...

For the orbit of a point $x$ in Teichmüller space under the mapping class group, show that as $r\to\infty$: (1) the proportion of orbit points in the ...