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Period-multiplying cascades for diffeomorphisms of the disc

Published online by Cambridge University Press:  24 October 2008

Jean-Marc Gambaudo ,
John Guaschi  and
Toby Hall
Jean-Marc Gambaudo
Affiliation:
Institut Non-Linéaire de Nice, Université de Nice Sophia Antipolis, Faculté des Sciences, 06108 Nice Cedex 2, France
John Guaschi
Affiliation:
Institut Non-Linéaire de Nice, Université de Nice Sophia Antipolis, Faculté des Sciences, 06108 Nice Cedex 2, France
Toby Hall
Affiliation:
Institut Non-Linéaire de Nice, Université de Nice Sophia Antipolis, Faculté des Sciences, 06108 Nice Cedex 2, France
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It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 359 - 374
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

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