A homological condition for a dynamical and illuminatory classification of torus branched coverings

Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of the moduli spaces of pairs (curve, holomorphic differential) with fixed multiplicities of zeros of the differential and have several applications in ergodic theory.

分支覆盖映射中的函数方程

Functional equations in branched covering maps

本文研究闭曲面间分支覆盖映射的函数方程，将半共轭全纯映射结果推广至更广情形，并给出环面自同态商具有抛物轨道流形的另证及推广。

Multi-branched resonances, chaos through quasiperiodicity, and asymmetric states in a superconducting dimer

A system of two identical SQUIDs (superconducting quantum interference devices) symmetrically coupled through their mutual inductance and driven by a sinusoidal field is investigated numerically with respect to dynamical properties such as its multibranched resonance curve, its bifurcation structure, as well as its synchronization behavior. The SQUID dimer is found to exhibit a hysteretic resonance curve with a bubble connected to it through Neimark-Sacker (torus) bifurcations, along with coexisting chaotic branches in their vicinity. Interestingly, the transition of the SQUID dimer to chaos o…

Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence

Let $\tilde f\colon(S^2,\tilde A)\to(S^2,\tilde A)$ be a Thurston map and let $M(\tilde f)$ be its mapping class biset: isotopy classes rel $\tilde A$ of maps obtained by pre- and post-composing $\tilde f$ by the mapping class group of $(S^2,\tilde A)$. Let $A\subseteq\tilde A$ be an $\tilde f$-invariant subset, and let $f\colon(S^2,A)\to(S^2,A)$ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group $\mathrm{Mod}(S^2,\tilde A)$ is an iterated extension of $\mathrm{Mod}(S^2,A)$ by fundamental groups of punctured spheres, $M(\tilde f)$ is an …

Algorithmic aspects of branched coverings

This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets. We introduce a canonical "Levy" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps. As an outcome, we pro…

Algorithmic aspects of branched coverings II/V. Sphere bisets and their decompositions

We consider "Thurston maps": branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as "mapping class bisets". We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called "small mapping class bisets". We phrase t…