Crinkled changes of variables for maps on a circle
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REVIEW
Crinkled changes of variables for maps on a circle Suddhasattwa Das · James A. Yorke
Received: 25 January 2020 / Accepted: 20 March 2020 © Springer Nature B.V. 2020
Abstract Take a piece of paper, crush it into a ball, and pound it flat onto a surface. The map from the original piece of paper to the surface is a non smooth change of variables Φ. It maps many points to the same point. This is what we mean by a “crinkled” map. Except that we work in one dimension, not two. Such crinkled change of variables occur naturally in many dynamical systems. One of the simplest dynamical systems with complicated dynamics is z n+1 = 2 · z n mod 1. A general nonlinear map on the circle such as xn+1 = 2·xn +sin(2π xn ) mod 1 has more complicated dynamics; in particular it has many more period N orbits for all N > 1. We show there is a continuous change of variables function z = Φ(x) that maps the circle onto itself. By z n = Φ(xn ), it converts the x trajectories into the simpler system z n+1 = m ·z n mod 1. This Φ serves as a change of variables, which in many cases will be shown to be non monotonic and nowhere differentiable. Our goal is to explain this sophisticated theory which is little known outside the technical dynamics literature. We hope that our examples and proofs will give the reader a better understanding of the outcomes of changes of variables. S. Das (B) Courant Institute of Mathematical Sciences, New York, USA e-mail: [email protected] J. A. Yorke Institute for Physical Sciences and Technology and the Departments of Math and Physics, University of Maryland College Park, College Park, USA
Keywords Semiconjugacy · Circle-map · Cantor set · Lifts
1 Introduction This paper is for a special issue of Dynamical Systems aimed at “Chaos theory …: A retrospective on lessons learned and missed ...” . We revisit some important results [1–3] from dynamical systems theory, which have not achieved a wide readership due to their generally technical nature and abstract presentation, a necessity due to their aim for generality. For example, Franks’ results [1] are formulated in terms of “infranil manifold endomorphisms”. We work in the more familiar case of a map F from the circle into itself, as does MacKay’s brief report [2] on this subject. But it is less than a page and also lacks figures. An important contribution of ours for the non specialist is our Figs.1, 2, 3, 4 and 5. They provide motivating and interesting illustrations of the results, particularly of the rich complexity of dynamics that may be contained in simple circle maps. In mathematics there are many investigations of a map Φ from one space Z onto another X , under the assumption that Φ maps some mathematical structure of Z maps onto that of X . We are interested in cases where the map is not one-to-one, such as for a map from a higher dimensional vector space onto a lower dimensional one—requiring that the linear structure is preserved, and we can ask about the inverse images
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S. Das, J. A. Yorke
Fig. 1 A Folding of a circle map onto a circle
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