Nonstationary normal forms for anosov diffeomorphisms and hyperbolic skew products

Citation

DeLatte, David A. (1991) Nonstationary normal forms for anosov diffeomorphisms and hyperbolic skew products. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xk38-8n02. https://resolver.caltech.edu/CaltechETD:etd-03292004-155247

Abstract

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The central theorems in my thesis are generalizations of theorems due to G.Birkhoff, S.Sternberg, and J.Moser on local normal forms for invertible mappings. We will consider smooth, area preserving, Anosov diffeomorphisms of the two dimensional torus, [...]. These are among the most fundamental examples of dynamical systems which exhibit extremely complicated (chaotic) behavior. Some geometric consequences and applications to rigidity phenomenon are also explored.

Let [...] be a smooth area preserving mapping for which the origin is a hyperbolic fixed point. Birkhoff considered the formal power series of f at the origin and showed that there is a formal change of coordinates, h, which satisfies [...] where g is of the form [...] and where [...] is the (real) eigenvalue of the linear part [...] and [...] is a formal series with [...]. The map g is called the local normal form for f. S.Sternberg showed that if the function f is [...] then h and hence g can be chosen to be [...] also. J. Moser was able to show that if f is analytic then Birkhoff's formal series for h and g converge in a small neighborhood of the origin. Note that the hyperbolae xy = constant are invariant curves for g. One may introduce hyperbolic coordinates [...] where [...] the hyperbolic angle, describes the position on the hyperbola xy = c. These coordinates give a clear understanding of the local behavior of f; specifically f shifts points along these local hyperbolae.

These theorems are generalized by eliminating the necessity of working at a fixed point for the map. Consider a smooth, area preserving, Anosov (i.e., hyperbolic at each point) diffeomorphism, f, of the two dimensional torus, [...]. There exists a family of local coordinate changes [...] which transform [...] (the local representation for f) into the normal form [...]. Furthermore [...] are continuous in p.

The first step of the proofs in both the [...] and analytic cases is to establish a nonstationary version of the formal theorem above. From the formal solution one can construct a [...] representation for [...] which is area preserving and satisfies the conjugacy equation above in a neighborhood of p. In the analytic case a majorization scheme is employed to demonstrate the convergence of [...]. One should also be able to use a rapidly converging iteration method instead of majorization. Our proofs do not fully exploit the fact that the manifold is the torus (compactness and two dimensionality are used). The theorem above holds if we replace the torus with a fibre bundle which has a compact base and two dimensional fibres and the mapping with a hyperbolic skew product transformation.

Since the hyperbolae xy = c are "preserved" by the nonstationary normal form, hyperbolic coordinates are available in this case also. The map [...] takes the simple form [...]. One can interpret [...] as a nonstationary hyperbolic twist. The higher order terms of the second coordinate of [...] form 1-cocycles (in the sense of group cohomology for [...] actions). Let [...] denote the first nonlinear term of the normal form for [...]. By integrating [...] over [...] one obtains a global invariant of the dynamical system [...].

Thesis Files