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Quasi-periodic orbits in Siegel disks/balls and the Babylonian problem. (English) Zbl 1412.37052

Summary: We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasi-periodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this “linearization” (or conjugacy) from knowledge of a single quasi-periodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian problem: determining the characteristics of a quasi-periodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method “the weighted Birkhoff average”.

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37A30 Ergodic theorems, spectral theory, Markov operators
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