Nielsen theory for periodic orbits and applications to dynamical systems. (English) Zbl 0798.55001
McCord, Christopher K. (ed.), Nielsen theory and dynamical systems. AMS- IMS-SIAM summer research conference, June 20-26, 1992, South Hadley, MA (USA), supported by the National Science Foundation. Providence, RI: American Mathematical Society. Contemp. Math. 152, 183-202 (1993).
This is an exposition of the results in the author’s ‘Estimation of the number of periodic orbits’ (Preprint), and some of his related work. It contains illustrative examples, but no proofs.
The author’s abstract: “A Nielsen theory for periodic orbits is presented that allows quantitative estimation of the number of periodic orbits of a self-map \(f: X\to X\) of a compact polyhedron. Of special interest is the estimation of the asymptotic growth rate of the number of periodic orbits and the homotopy invariant lower estimation of the topological entropy, via a matrix representation of \(\pi_ 1(T_ f)\), where \(T_ f\) is the mapping torus of \(f\). As applications, some recent results in two-dimensional dynamical systems theory are improved”.
For the entire collection see [Zbl 0780.00035].
The author’s abstract: “A Nielsen theory for periodic orbits is presented that allows quantitative estimation of the number of periodic orbits of a self-map \(f: X\to X\) of a compact polyhedron. Of special interest is the estimation of the asymptotic growth rate of the number of periodic orbits and the homotopy invariant lower estimation of the topological entropy, via a matrix representation of \(\pi_ 1(T_ f)\), where \(T_ f\) is the mapping torus of \(f\). As applications, some recent results in two-dimensional dynamical systems theory are improved”.
For the entire collection see [Zbl 0780.00035].
Reviewer: H.Schlagbauer (Windhoek)
MSC:
| 55M20 | Fixed points and coincidences in algebraic topology |
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
| 58C30 | Fixed-point theorems on manifolds |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
