A HORSESHOE WITH POSITIVE MEASURE
DOI:
https://doi.org/10.5902/2179460X12131Keywords:
Dynamical systems, Smale’s horseshoe, Cantor set.Abstract
http://dx.doi.org/10.5902/2179460X12131
The present paper aims to establish the bases for the construction of the Smale’s Horseshoe with positive measure, originallypresented by Rufus Bowen in an article from 1975. To attain this, we start with the construction of Stephen Smale’s classicalhorseshoe, presented by Smale in 1967. Considering figure D in the shape of a stadium on the plane, which contains one square,Q, we will make one contraction and one expansion of D to obtain a figure which looks like a horseshoe. The operations ofcontracting and expanding define one function, that is, f : D ! D. We define Smale’s horseshoe as set l of points q 2 Q Dwhich belong to Q after future and past interactions of function f . This set L is a Cantor set. Bowen’s horseshoe, i.e., Smale’shorseshoe of positive measure, is a set which duplicates Smale’s horseshoe.
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