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Small Divisors: Number Theory in Dynamical Systems

In: An Invitation to Mathematics

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  • Jean-Christophe Yoccoz

    (Collège de France)

Abstract

We discuss dynamical systems with two or more moving particles, such as two planets orbiting around the sun. If the ratio of their rotation periods, say α, is rational, then the planets are in resonance, and the mutual interaction will make the dynamics unstable. If the period ratio α is irrational, it can be approximated arbitrarily well by rational numbers, and the stability depends on how good this approximation is in terms of the sizes of numerators and denominators. We discuss this in a mathematical model case that can be analyzed completely, the setting of iteration of quadratic polynomials z↦e 2πiα z+z 2, and show how this leads to questions of Diophantine approximation within number theory. Finally, we briefly mention the situation of more than two planets.

Suggested Citation

  • Jean-Christophe Yoccoz, 2011. "Small Divisors: Number Theory in Dynamical Systems," Springer Books, in: Dierk Schleicher & Malte Lackmann (ed.), An Invitation to Mathematics, pages 43-54, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-19533-4_4
    DOI: 10.1007/978-3-642-19533-4_4
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