The main theme of this
paper is to estimate, for self-maps f : X → X of compact polyhedra, the
asymptotic Nielsen number N∞(f) which is defined to be the growth rate
of the sequence {N(fn)} of the Nielsen numbers of the iterates of f. 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.
