Circle maps: Irrationally winding. (English) Zbl 0796.58022
Waldschmidt, Michel (ed.) et al., From number theory to physics. Lectures of a meeting on number theory and physics held at the Centre de Physique, Les Houches (France), March 7-16, 1989. Berlin: Springer-Verlag. 631-658 (1992).
In this survey, circle maps are discussed as an example of a physically interesting chaotic dynamical system with rich number-theoretic structure.
Three distinct thermodynamic formulations are discussed: the Farey series, the continued fractions of fixed length and the Farey tree levels. The results of the circle-map renormalization theory are briefly summarized and the relation of Farey series to deep problems in number theory, such as the Riemann hypothesis, is described. But the main topic is the study of universal properties of the entire irrational winding set such as the universality, indicated by the numerical work, of the Hausdorff dimension of the set of irrational windings for critical circle maps with cubic inflection.
For the entire collection see [Zbl 0784.00021].
Three distinct thermodynamic formulations are discussed: the Farey series, the continued fractions of fixed length and the Farey tree levels. The results of the circle-map renormalization theory are briefly summarized and the relation of Farey series to deep problems in number theory, such as the Riemann hypothesis, is described. But the main topic is the study of universal properties of the entire irrational winding set such as the universality, indicated by the numerical work, of the Hausdorff dimension of the set of irrational windings for critical circle maps with cubic inflection.
For the entire collection see [Zbl 0784.00021].
Reviewer: Ľ.Snoha (Banská Bystrica)
MSC:
| 37B99 | Topological dynamics |
| 37A99 | Ergodic theory |
| 54H20 | Topological dynamics (MSC2010) |
| 26A18 | Iteration of real functions in one variable |
| 11B57 | Farey sequences; the sequences \(1^k, 2^k, \dots\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
