More Like this

1. The Kodaira dimension of complex hyperbolic manifolds with cusps

   https://doi.org/10.1112/S0010437X1700762X  

   Bakker, Benjamin; Tsimerman, Jacob (March 2018, Compositio Mathematica)

   We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $$n$$ -dimensional toroidal compactification $$\overline{X}$$ with boundary $$D$$ , $$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$$ is ample for $$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$$ , and in particular that $$K_{\overline{X}}$$ is ample for $$n\geqslant 6$$ . By an independent algebraic argument, we prove that every ball quotient of dimension $$n\geqslant 4$$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture. 

   more » « less
2. Invariant random subgroups of semidirect products

   https://doi.org/10.1017/etds.2018.46  

   BIRINGER, IAN; BOWEN, LEWIS; TAMUZ, OMER (January 2018, Ergodic Theory and Dynamical Systems)

   We study invariant random subgroups (IRSs) of semidirect products $$G=A\rtimes \unicode[STIX]{x1D6E4}$$ . In particular, we characterize all IRSs of parabolic subgroups of $$\text{SL}_{d}(\mathbb{R})$$ , and show that all ergodic IRSs of $$\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$$ are either of the form $$\mathbb{R}^{d}\rtimes K$$ for some IRS of $$\text{SL}_{d}(\mathbb{R})$$ , or are induced from IRSs of $$\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$$ , where $$\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$$ is a lattice. 

   more » « less
3. Dynamical degrees of Hurwitz correspondences

   https://doi.org/10.1017/etds.2018.125  

   RAMADAS, ROHINI (July 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$\unicode[STIX]{x1D719}$$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $$\unicode[STIX]{x1D719}$$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence $${\mathcal{H}}_{\unicode[STIX]{x1D719}}$$ —of the moduli space $${\mathcal{M}}_{0,\mathbf{P}}$$ . We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees . We show that the sequence of dynamical degrees of $${\mathcal{H}}_{\unicode[STIX]{x1D719}}$$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $$\unicode[STIX]{x1D719}$$ at and near points of its post-critical set. 

   more » « less
4. On the computability of rotation sets and their entropies

   https://doi.org/10.1017/etds.2018.45  

   BURR, MICHAEL A.; SCHMOLL, MARTIN; WOLF, CHRISTIAN (February 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$f:X\rightarrow X$$ be a continuous dynamical system on a compact metric space $$X$$ and let $$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$$ be an $$m$$ -dimensional continuous potential. The (generalized) rotation set $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is defined as the set of all $$\unicode[STIX]{x1D707}$$ -integrals of $$\unicode[STIX]{x1D6F7}$$ , where $$\unicode[STIX]{x1D707}$$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $$\unicode[STIX]{x210B}(w)$$ to each $$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $$f$$ is a subshift of finite type. We prove that $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is computable and that $$\unicode[STIX]{x210B}(w)$$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $$\unicode[STIX]{x210B}$$ is not continuous on the boundary of the rotation set when considered as a function of $$\unicode[STIX]{x1D6F7}$$ and $$w$$ . In particular, $$\unicode[STIX]{x210B}$$ is, in general, not computable at the boundary of $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ . 

   more » « less
5. Rotation of a superhydrophobic cylinder in a viscous liquid

   https://doi.org/10.1017/jfm.2019.776  

   Yariv, Ehud; Siegel, Michael (December 2019, Journal of Fluid Mechanics)

   The hydrodynamic quantification of superhydrophobic slipperiness has traditionally employed two canonical problems – namely, shear flow about a single surface and pressure-driven channel flow. We here advocate the use of a new class of canonical problems, defined by the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally measured by the enhancement of the Stokes mobility relative to the corresponding mobility of a homogeneous particle. We focus upon what may be the simplest problem in that class – the rotation of an infinite circular cylinder whose boundary is periodically decorated by a finite number of infinite grooves – with the goal of calculating the rotational mobility (velocity-to-torque ratio). The associated two-dimensional flow problem is defined by two geometric parameters – namely, the number $$N$$ of grooves and the solid fraction $$\unicode[STIX]{x1D719}$$ . Using matched asymptotic expansions we analyse the large- $$N$$ limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation, $$\begin{eqnarray}\displaystyle 1+{\displaystyle \frac{2}{N}}\ln \csc {\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D719}}{2}}, & & \displaystyle \nonumber\end{eqnarray}$$ for the ratio of the enhanced mobility to the homogeneous-cylinder mobility. Making use of conformal-mapping techniques and inductive arguments we prove that the preceding approximation is actually exact for $$N=1,2,4,8,\ldots$$ . We conjecture that it is exact for all $$N$$ . 

   more » « less