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

Humorous Field - Field to Humour = 幽默地誌 - 地致幽默

This is an experiment in spatial humour, and this assumption of humour can be applied to any urban space. Since this project starts with Lingnan University’s support of its Artist-in-Residence Program, it would seem appropriate to base the experiment around the university campus. Two straight lines were laid on the map of Tuen Mun. The lines extend to East-West through the ridge. The line to the South stretches across the valley and ends on the South coast. Two transects span a wide variety spaces and urban life in Tuen Mun. They would become the premises of this experiment. Exploring the two fixed paths, the artist observes and discovers potential humour along the route. This methodology was adapted from census data collection of marine life, in which marine biologists follow a selected transect across the sea floor and record the species of plants and sea animal encountered. 這是一個關於「空間幽默」的實驗，場所可以是任意的城市空間。 感謝嶺南大學給予藝術家駐校的機會。此實驗以大學為中心，在地圖上劃出縱橫線。縱線由北面泥圍起，途經大學、新墟街市、屯門河、蝴蝶邨等，伸延至南面海岸終。橫線由東面山脊始，途經虎地、兆康西鐵站、小坑村等，到西面山脊止。兩條直線貫穿之地大致包含了屯門社區各種空間及社群。藝術家遊走於這兩條指定路線，並嘗試運用自己的觀察及想像，發掘社區中的空間幽默。（此實驗方式效法海洋生態普查實驗：在海底拉出一條五十米長直绿，觀察和記錄所經之處的魚類品種及大小）。https://commons.ln.edu.hk/vs_artist_catalog/1019/thumbnail.jp