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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A pathological area preserving $ C\sp{\infty }$ diffeomorphism of the plane


Author: Michael Handel
Journal: Proc. Amer. Math. Soc. 86 (1982), 163-168
MSC: Primary 58E99; Secondary 28D05, 54F20, 58F12
MathSciNet review: 663889
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Abstract: The pseudocircle $ P$ is an hereditarily indecomposable planar continuum. In particular, it is connected but nowhere locally connected. We construct a $ {C^\infty }$ area preserving diffeomorphism of the plane with $ P$ as a minimal set. The diffeomorphism $ f$ is constructed as an explicit limit of diffeomorphisms conjugate to rotations about the origin. There is a well-defined irrational rotation number for $ f\vert P$ even though $ f\vert P$ is not even semi-conjugate to a rotation of $ {S^1}$. If we remove the requirement that our diffeomorphisms be area preserving, we may alter the example so that $ P$ is an attracting minimal set.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0663889-6
PII: S 0002-9939(1982)0663889-6
Article copyright: © Copyright 1982 American Mathematical Society



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