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Generalized Lyapunov exponents corresponding to higher derivatives. (English) Zbl 0778.58036
Let \(f:I\to I\) be a differentiable map on some \(I\subset\mathbb{R}\). The paper introduces the expression \(\lim_{n\to\infty}{1\over n}\log|{d^ p\over dx^ p}f^ n(x)|\) to be the \(p\)th order Lyapunov exponent. The notion is extended to \(f:M\to M\), \(M\) an \(n\)-dimensional manifold, expressing the \(p\)th derivative of \(f\) by the \(p\)th derivative of the induced map in a chart. The cases of fixed points and of periodic orbits of \(f\) are discussed in some detail for both the one- and the \(n\)- dimensional case, providing expressions for the \(p\)th order exponents in terms of the first order exponents.

MSC:
37A99 Ergodic theory
37E99 Low-dimensional dynamical systems
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