arXiv · math/0411343
Informal seminar notes explaining the ideas in math.DS/0411344 and math.DS/0411345.
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Tom Leinster. 2004-11-15. General self-similarity: an overview. https://arxiv.org/abs/math/0411343
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We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in its own right, is a combinatorial classification of critically fixed anti-rational maps as "Schottky maps" associated with certain plane graphs. Both of these classification results heavily rely on an orientation-reversing version of Thurston's theory, including the canonical decomposition of anti-Thurston maps, which we develop in this paper. We use our combinatorial framework to answer special instances of the global curve attractor and twisting problems, and to resolve the Hurwitz existence problem in the setting of critically fixed anti-rational maps. Finally, we give some applications to the study of anti-rational maps with symmetries and to critically fixed anti-polynomials.
We investigate the strong stable sets of the geodesic flow on hyperbolic surfaces and their relation to the injectivity radius. We prove that the strong stable set of a vector does not coincide with its stable horocycle if and only if the injectivity radius along the associated geodesic ray is not bounded away from zero and the ray does not diverge through a cusp. In that case, the strong stable set is an uncountable union of stable horocycles. In previous works, we considered the particular cases where the ray meets arbitrarily short closed geodesics or closed horocycles. Here we complete the study by treating the third possible case, where the ray meets closed hypercycles of arbitrarily small length.
Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.