Abstract
A recent paper considers the dependence of the size of analyticity domains of some functions appearing in KAM theory as a function of the distance to breakdown. They tentatively conclude that the relation is linear. In this note we argue that McKay's renormalization group picture predicts a power-law dependence with an exponent close to 1 but not equal to 1.
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References
A. Berretti, A. Celletti, L. Chierchia, and C. Falcolini, Natural boundaries for area preserving twist maps.J. Stat. Phys. 66:1613–1630 (1992).
J. Wilbrink, Erratic behaviour of invariant circles in standard-like mappings.Physica 26D:358–368 (1897).
J. Wilbrink, New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps.Nonlinearity 3:567–584 (1990).
R. S. MacKay, A renormalisation approach to invariant circles in area preserving maps.Physica 7D:283–300 (1983).
J. A. Ketoja and R. S. MacKay, Fractal boundary for the existence of invariant circles for area preserving maps: observations and a renormalisation explanation.Physica 35D:318–334 (1989).
C. Falcolini and R. de la Llave, Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors.J. Stat. Phys., to appear.
R. S. MacKay, Renormalisation in area preserving maps. Princeton thesis (1982).
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de la Llave, R. A renormalization group explanation of numerical observations of analyticity domains. J Stat Phys 66, 1631–1634 (1992). https://doi.org/10.1007/BF01054438
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DOI: https://doi.org/10.1007/BF01054438