8 problems

Let $\Omega\subset\mathbb{R}^{n}$ be a domain, and let $f\in W_{loc}^{1,n}(\Omega,\mathbb{R}^{n})$ be a Sobolev map with distortion function $K\in L_{loc}^{n-1}(\Omega)$. Iwaniec–Š…

Bonk–Heinonen conjecture. The bound can be chosen independently of $K$; equivalently, there should be a constant depending only on $d$ and $l$ such that

Let $n\geq 3$, and let $K_0=K_0(n)>1$ be a constant such that every $K$-quasiregular map $f\colon\Omega\to\mathbb{R}^n$ with $K<K_0$ is a local homeomorphism. The winding map is th…

Martio's conjecture. If $n\geq3$ and $K_I(f)<2$, then

Positive-capacity conjecture. If the Julia set is infinite, then it has positive capacity, so the positive-capacity assumption is unnecessary for the equality…

Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold with $n\geq 2$, and suppose that $M$ is not a rational cohomology sphere. A self-map $f:M\to M$ is uniformly qu…

Let $M$ be a closed oriented Riemannian $n$-manifold, let $f\colon\mathbb{R}^n\to M$ be quasiregular, and let $\Gamma$ be a discrete subgroup of the isometry group $E(n)$ with tran…

Martin–Mayer's conjecture. If the Julia set has positive measure, then $f$ is of Lattès type: