A curious dynamical system in the plane

2024/09/13 by Steinerberger, Stefan, Zeng, Tony · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.08961

Abstract

For any irrational α> 0 and any initial value z_-1 ∈ ℂ, we define a sequence of complex numbers (z_n)_n=0^∞ as follows: z_n is z_n-1 + e^{2 πi αn} or z_n-1 - e^{2 πi αn}, whichever has the smaller absolute value. If both numbers have the same absolute value, the sequence terminates at z_n-1 but this happens rarely. This dynamical system has astonishingly intricate behavior: the choice of signs in z_n-1 ± e^{2 πi αn} appears to eventually become periodic (though the period can be large). We prove that if one observes periodic signs for a sufficiently long time (depending on z_-1, α), the signs remain periodic for all time. The surprising complexity of the system is illustrated through examples.

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