Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior

This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cusps. These minimizers are the same ones identified by Hironaka, Aaber-Dunfiled, Kin-Takasawa independently. In the case $g \equiv 6 \pmod{12}$ we find a new family of pseudo-Anosovs defined on $\varSigma_g$ with orientable invariant foliations obtained from N(-6) or N(4) by Dehn filling two cusps. We prove that if $\delta_g^+$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on $\varSigma_g$, then $$\limsup_{\substack{g \equiv 6 \pmod{12} g \to \infty}} g \log \delta^+_g \le 2 \log \delta(D_5) \approx 1.0870,$$ where $\delta(D_n)$ is the minimal dilatation of pseudo-Anosovs on an $n$-punctured disk. We also study monodromies of fibrations on N(1). We prove that if $\delta_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with $n$ punctures, then $$\limsup_{n \to \infty} n \log \delta_{1,n} \le 2 \log \delta(D_4) \approx 1.6628.$$Comment: 46 pages, 14 figures; version 3: Major change in Section 2.1, and minor correction

City of Lions

In the course of the past two decades, the city of Lviv has enjoyed close attention as well as a “close reading” in literary and scholarly texts on the city. This attention fits easily into two categories: (a) scholars producing academic studies on the city and (b) classical literary works on the city, composed in various languages, finally becoming available to a broader readership through translation into English. The book under discussion falls into the second category.1 It must be pointed out right away that this is an unusual book—a truly successful combination of two essays—that should be rewarded with proper attention. Under one cover, the reader has the opportunity to enjoy two pieces that are linked together by the image of the city of Lviv. The first is an elegiac essay, Mój Lwów (My Lviv), authored by Polish writer Józef Wittlin. (This first English-language translation is by Antonina Lloyd-Jones.) The second essay is from post‑2010s Lviv by internationally-recognized British author Philippe Sands; it uses Wittlin’s work as a springboard for Sands’ own explorations of the city, or of what is left of the city from that period, mixed with a personal narrative

The forcing partial order on a family of braids forced by pseudo-Anosov 3-braids

Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland's dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map which is a factor possessing complicated chaotic dynamics.Comment: 16 pages, 12 figure