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Geometric expansion, Lyapunov exponents and foliations. (English) Zbl 1173.37031
Let \(M\) be an \(n\)-dimensional compact Riemannian manifold, and let \(f \in \text{Diff}^r(M)\) be a \(C^r\) diffeomorphism on \(M\) \((r \geq 1)\). The growth rate of a uniformly expanding foliation, whose examples can be found in hyperbolic and partially hyperbolic diffeomorphisms on \(M\), is measured in several different ways. In this paper, the authors define three types of growth rates for the foliation under \(f\), and discuss some properties and show relationships among them. The geometric growth rate for the foliation is first defined in relation to the volume growth used by Y. Yomdin [Isr. J. Math. 57, 285–300 (1987; Zbl 0641.54036)] and S. E. Newhouse [Ann. Math. (2) 129, No. 2, 215–235 (1989; Zbl 0676.58039)]. The topological growth rate for the foliation depends on the homology that the invariant foliation carries and on the action induced by the map on the homology. Typically the topological growth will be much more easier to compute than the geometric one and it is a local constant for maps in \(\text{Diff}^r(M)\). It is shown that the geometric growth and the topological growth are the same for foliations carrying certain homological information. The third type of growth rate which is called the Lyapunov growth is measured by the Lyapunov exponents in the tangent spaces of the leaves of the foliations. It is shown that if the foliation is absolutely continuous, then the Lyapunov growth is smaller than the geometric growth. As an application, the authors show examples of (volume-preserving) systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of M. Shub and A. Wilkinson [Invent. Math. 139, No. 3, 495–508 (2000; Zbl 0976.37013)] to cases where the center manifolds are not compact.

MSC:
37D30 Partially hyperbolic systems and dominated splittings
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
26B15 Integration of real functions of several variables: length, area, volume
57R30 Foliations in differential topology; geometric theory
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References:
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