An Anosov automorphism on R2 is a mapping from the unit square S onto S of the form x y

Chapter 10, Problem 6

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An Anosov automorphism on R2 is a mapping from the unit square S onto S of the form x y a b c d x y mod 1 in which (i) a, b, c, and d are integers, (ii) the determinant of the matrix is 1, and (iii) the eigenvalues of the matrix do not have magnitude 1. It can be shown that all Anosov automorphisms are chaotic mappings. (a) Show that Arnolds cat map is an Anosov automorphism. (b) Which of the following are the matrices of an Anosov automorphism? 0 1 1 0 , 3 2 1 1 , 1 0 0 1 , 5 7 2 3 , 6 2 5 2 (c) Show that the following mapping of S onto S is not an Anosov automorphism. x y 0 1 1 0 x y mod 1 What is the geometric effect of this transformation on S? Use your observation to show that the mapping is not a chaotic mapping by showing that all points in S are periodic points.