Let M be a compact
manifold and f : M → M a self map on M. For any natural number n, the n-th
iterate of f is the n-fold composition fn: M → M. The fixed point set of f is
fix(f) = {x ∈ M : f(x) = x}. We say that x ∈ M is a periodic point of f if x is a
fixed point of some fn and we denote the set of all periodic points of f by
per(f) =⋃n=1∞fix(fn). A periodic point x of f is said to have minimal period k
provided that k is the smallest integer for which x ∈fix(fk). We say that per(f)
is homotopically finite, denoted per(f) ∼ finite, iff there is a g homotopic
to f such that per(g) is finite. When M is a torus B. Halpern has shown
that per(f) ∼ finite iff the sequence of Nielsen numbers {N(fn)}n=1∞ is
bounded. The main objective of this work is to extend these results to all
nilmanifolds and to consider to what extent they can be extended for compact
solvmanifolds. A compact nilmanifold is a coset space of the form M = G∕Γ
where G is a connected, simply connected nilpotent Lie group and Γ is a
discrete torsion free uniform subgroup. For these spaces we have the additional
result that when the homotopy can be accomplished, the resulting g satisfies
|fix(gn)| = N(fn) for all n with N(fn)≠0 and if {N(fn)}n=1∞= {0} then we can
choose g to be periodic point free. Also, when f is induced by a homomorphism
F : G → G, then we can write g = ufv where u and v are isotopic to the
identity. This form for the homotopy is used to find sufficient conditions for
per(f) ∼ finite when M is a solvmanifold. We then present a model for a
specific class of solvmanifolds where these conditions can be considered.
This allows us to prove the general result in a variety of low dimensional
examples.
