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Perturbing the Cat Map: Mixed Elliptic and Hyperbolic Dynamics

Applied Mathematics,Ìý

Date and time:Ìý

Thursday, February 5, 2015 - 2:00pm

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ECCR 257

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Arnold’s cat map is a prototypical dynamical system on the torus with uniformly hyperbolic dynamics. Since the famous picture of a scrambled cat in the 1968 book by Arnold and Avez, it has become one of the icons of chaos. In 2010, Lev Lerman studied a family of maps homotopic to the cat map that has, in addition to a saddle, a parabolic fixed point. Lerman conjectured that this map could be a prototype for dynamics with a mixed phase space, having positive measure sets of nonuniformly hyperbolic and of elliptic orbits. We present some numerical evidence that supports Lerman’s conjecture. The elliptic orbits appear to be confined to a pair of channels bounded by invariant manifolds of the two fixed points. The complement of the channels appears to be a positive measure Cantor set. Computations show that orbits in the complement have positive Lyapunov exponents.