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< BR > < p对齐= "证明" >我的同构问题n dynamics dates back to a question of von Neumann from 1932: Is it possible to classify (in some reasonable sense) the ergodic measure-preserving diffeomorphisms of a compact manifold up to isomorphism? We want to study a similar problem: Let C be the Cantor set and let Min(C) stand for the space of all minimal homeomorphisms of the Cantor set. Recall that a homeomorphism f is in Min(C) if every orbit of f is dense in C. We say that f and g in Min(C) are topologically conjugate if there is a Cantor set homeomorphism h such that f o h = h o g. We prove an anti-classification result showing that even for very liberal interpretations of what a "reasonable'' classification scheme might be, a classification of minimal Cantor set homeomorphism up to topological conjugacy is impossible. We see it as a consequence of the following: we prove that the topological conjugacy relation of Cantor minimal systems TopConj treated as a subset of Min(C)xMin(C) is complete analytic. In particular, TopConj is a non-Borel subset of Min(C)xMin(C). Roughly speaking, it is impossible to tell if two minimal Cantor set homeomorphisms are topologically conjugate using only a countable amount of information and computation.

Our result is proved by applying a Foreman-Rudolph-Weiss-type construction used for an anti-classification theorem for ergodic automorphisms of the Lebesgue space. We find a continuous map F from the space of all trees over non-negative integers with arbitrarily long branches into the class of minimal homeomorphisms of the Cantor set. Furthermore, F is a reduction, which means that a tree T is ill-founded if and only if F(T) is topologically conjugate to its inverse. Since the set of ill-founded trees is a well-known example of a complete analytic set, it is impossible to classify which minimal Cantor set homeomorphisms are topologically conjugate to their inverses.

This is joint work with Konrad Deka, Felipe García-Ramos, Kosma Kasprzak, Philipp Kunde (all from the Jagiellonian University in Kraków).

可以递归地计算通过阶乘ir definition, but the computation gets difficult quite quickly when the number of interest gets larger and larger. A workaround is given by Stirling’s approximation: a closed, asymptotic formula for factorials. A similar (but much more complicated) situation occurs when trying to compute Witten’s intersection numbers. Virasoro constraints recursively compute these numbers, but the computation gets difficult when the genus gets larger and larger. An approximation has been recently proved by Aggarwal by studying the structure of the associated Virasoro constraints. I will present an alternative proof of Aggarwal’s result based on quantum curves and resurgence. The advantage of this strategy is that it easily generalises to several problems (like r-spin intersection numbers, Norbury’s intersection numbers, etc) and gives higher-order corrections.

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