Mathematics Problem Archive

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....

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...

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...

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...

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...

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....

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 ...