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Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. (English) Zbl 0763.58024
The objects of study in this paper are unimodular interval maps \(T:[0,1]\to[0,1]\), assumed to be \(C^ 3\) and to satisfy the so-called Collet-Eckmann condition \(\liminf_{n\to\infty}(DT^ n(Tc))^{1/n}\), where \(c\) is the assumed unique non-degenerate critical point of \(T\) (of order 1). For such maps and assuming additionally that \(T\) has an invariant probability density, two main aspects are investigated: mixing properties and the distribution of typical periodic orbits.
Using properties of the Perron Frobenius operator (quasi-compactness) as well as analytic properties of the zeta function an exponential mixing property is proven and a description is given of the long term behaviour of generic trajectories.

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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