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Fibre techniques in nielsen periodic point theory on solmanifolds III: Calculations
Abstract
This third paper of the series gives the
necessarily lengthy illustrations of the main results of the first two, delayed
until now for reasons of space. The illustrations in question concern
calculations of the Nielsen type periodic point numbers NPn And
NΦn(f) for self maps f of
solvmanifolds.
We
indicate
that for low dimensional solvmanifolds,
we
can often give formulae (as opposed to algorithms) for these numbers, which
of course include formulae for the
ordinary Nielsen numbers N(fn). We give a complete
analysis of all maps on two very different example generalizations of the Klein
bottle
K2. Both examples admit non-weakly Jiang maps, which is where the more complex
calculations occur. Our methods employ matrix theory and modular arguments with
periodic
matrices.
Among other things, our results include the promised
completion of the more difficult calculations on K2 itself,
as
well as a generalization of a surprising result first observed in Nielsen
periodic point theory on K2. More precisely we show for
each
m 〉 1 that
there is a solvmanifold Sof dimension m,
and self map f on S for which, NPn(f) =
N(fn) for an infinite number of n. We
also discuss the generality of our considerations, indicating that the type of
map
and solvmanifold
given here and the methods used, provide the data needed to make calculations
for all maps on all solvmanifolds. Finally we
indicate that data that allow the exhibited modular patterns, though widely
available, do not hold universally.
Mathematics Subject
Classification (2000):
55M20.
Key words: Nielsen numbers; periodic points; nilmanifolds; solvmanifolds.
Quaestiones
Mathematicae 25 (2002), 177-208
necessarily lengthy illustrations of the main results of the first two, delayed
until now for reasons of space. The illustrations in question concern
calculations of the Nielsen type periodic point numbers NPn And
NΦn(f) for self maps f of
solvmanifolds.
We
indicate
that for low dimensional solvmanifolds,
we
can often give formulae (as opposed to algorithms) for these numbers, which
of course include formulae for the
ordinary Nielsen numbers N(fn). We give a complete
analysis of all maps on two very different example generalizations of the Klein
bottle
K2. Both examples admit non-weakly Jiang maps, which is where the more complex
calculations occur. Our methods employ matrix theory and modular arguments with
periodic
matrices.
Among other things, our results include the promised
completion of the more difficult calculations on K2 itself,
as
well as a generalization of a surprising result first observed in Nielsen
periodic point theory on K2. More precisely we show for
each
m 〉 1 that
there is a solvmanifold Sof dimension m,
and self map f on S for which, NPn(f) =
N(fn) for an infinite number of n. We
also discuss the generality of our considerations, indicating that the type of
map
and solvmanifold
given here and the methods used, provide the data needed to make calculations
for all maps on all solvmanifolds. Finally we
indicate that data that allow the exhibited modular patterns, though widely
available, do not hold universally.
Mathematics Subject
Classification (2000):
55M20.
Key words: Nielsen numbers; periodic points; nilmanifolds; solvmanifolds.
Quaestiones
Mathematicae 25 (2002), 177-208
