The main theme of this paper is to estimate, for self-maps / : X -> X of compact polyhedra, the asymptotic Nielsen number 7V°° (/) which is defined to be the growth rate of the sequence {N(f n )} of the Nielsen numbers of the iterates of/. The asymptotic Nielsen number provides a homotopy invariant lower bound to the topological entropy h(f). To introduce our main tool, the Lefschetz zeta function, we develop the Nielsen theory of periodic orbits. Compared to the existing Nielsen theory of periodic points, it features the mapping torus approach, thus brings deeper geometric insight and simpler algebraic formulation. The important cases of homeomorphisms of surfaces and punctured surfaces are analysed. Examples show that the computation involved is straightforward and feasible. Applications to dynamics, including improvements of several results in the recent literature, demonstrate the usefulness of the asymptotic Nielsen number.Introduction.
The forcing relation of braids has been introduced for a 2-dimensional
analogue of the Sharkovskii order on periods for maps of the interval. In this
paper, by making use of the Nielsen fixed point theory and a representation of
braid groups, we deduce a trace formula for the computation of the forcing
order.Comment: 24 pages, 9 figure
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