Description and goals:
This course will study in some depth two distinct but related topics: The dynamics of (smooth) homeomorphisms of the circle, and the structure of rotation sets under the multiplication by $d$ map $t \mapsto d \, t$ (mod $\mathbb Z$) for some integer $d \geq 2$. Both theories arise in a variety of problems in one-dimensional complex dynamics, and occasionally merge. An illustrative example is the dynamics on the boundary of a Siegel disk of a polynomial of degree $d$ as seen from inside (irrational rotation) versus outside (multiplication by $d$). However, the results and tools that we discuss are general and go far beyond such particular examples.
The course will consist of 14 two-hour lectures. For each topic we start from the basics and take a path that gets us to interesting results as quickly as possible.
Here is a rough outline of possible topics:
Part I (about 9 lectures):
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Continued fraction algorithm, arithmetic of rigid rotations, combinatorial and statistical properties of their orbits.
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Rotation number, the Poincaré semiconjugacy, Denjoy's theorem on linearization of circle diffeomorphisms.
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Cross-ratio distortion, Świątek's cross-ratio inequality for critical circle maps.
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Herman-Świątek's real a priori bounds, linearization theorems of Yoccoz and Herman.
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Applications in constructing Siegel disks via surgery.
Part II (about 5 lectures):
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Rotation sets under $t \mapsto d \, t$ (mod $\mathbb Z$), minor and major gaps, gap dynamics, maximal and minimal rotation sets.
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The invariant and gap measures associated with a rotation set.
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New unified approach to the Goldberg-Milnor-Tresser deployment theorem.
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Low-degree computations: Rotation sets under the doubling and tripling maps.
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Connections with the dynamic planes and parameter spaces of families of complex polynomial maps.
General prerequisite: A good knowledge of graduate-level real and complex analysis and curiosity to learn about a beautiful piece of modern mathematics. Some modest background in dynamics is helpful, but much of the material will be accessible to anyone motivated enough to learn it.
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References