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Anosov Map


The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism, it is a map. In particular, an Anosov map is a C^1 map f of a manifold M to itself such that the tangent bundle of M is hyperbolic with respect to f.

A trivial example is to map all of M to a single point of M. Here, the eigenvalues are all zero. A less trivial example is an expanding map on the circle S^1, e.g., x|->2x (mod 1), where S^1 is identified with the real numbers (mod 1). Here, all the eigenvalues equal 2 (i.e., the eigenvalue at each point of S^1). Note that this map is not a diffeomorphism because f(x+(1/2))=f(x), so it has no inverse.

A nontrivial example is formed by taking Arnold's cat map on the 2-torus T^2, and crossing it with an expanding map on S^1 to form an Anosov map on the 3-torus T^3=T^2×S^1, where × denotes the Cartesian product. In other words,

 [x_(n+1); y_(n+1); z_(n+1)]=[1 1 0; 1 2 0; 0 0 2][x_n; y_n; z_n]     (mod 1).

See also

Anosov Diffeomorphism, Anosov Flow, Arnold's Cat Map

This entry contributed by Jonathan Sondow (author's link)

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References

Anosov, D. "Roughness of Geodesic Flows on Compact Riemannian Manifolds of Negative Curvature." Dokl. Akad. Nauk SSSR 145, 707-709, 1962. English translation in Soviet Math. Dokl. 3, 1068-1069, 1962.Anosov, D. "Ergodic Properties of Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature." Dokl. Akad. Nauk SSSR 151, 1250-1252, 1963. English translated in Soviet Math. Dokl. 4, 1153-1156, 1963.Lichtenberg, A. J. and Lieberman, M. A. Regular and Chaotic Dynamics, 2nd ed. New York: Springer-Verlag, pp. 305-307, 1992.Sondow, J. "Fixed Points of Anosov Maps of Certain Manifolds." Proc. Amer. Math. Soc. 61, 381-384, 1976.

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Anosov Map

Cite this as:

Sondow, Jonathan. "Anosov Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AnosovMap.html

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