Alexander Fel'shtyn
Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion
Preprint series: ESI preprints

MSC:

58-02 Research exposition (monographs, survey articles)

58F99 None of the above but in this section

58G99 None of the above but in this section

19J10 Whitehead (and related) torsion

Abstract: The article consists of four parts. Part I( Chapter 1)
presents a brief account of the Nielsen fixed point theory.
Part II( Chapters 2 - 4) deals with dynamical zeta functions connected
with Nielsen fixed point theory. Part III ( Chapter 5) is concerned with
congruences for the Reidemeister and Nielsen numbers. Part IV (Chapter 6)
deals with the Reidemeister torsion .

In Chapter 1 we define the lifting and fixed point classes, fixed point
index, Reidemeister and Nielsen numbers. The relevant definitions and results
will be used throughout the book.

In Chapter 2 - 4 we introduce the Reidemeister zeta functions of a group
endomorphism and of a map and the Nielsen zeta function of a map which are
the main objects of the monograph.

In Chapter 2 we prove that the Reidemeister zeta function of a group
endomorphism is a rational function with functional equation in the
following cases: the group is finitely generated and an endomorphism is
eventually commutative; the group is finite ; the group is a direct sum
of a finite group and a finitely generated free abelian group; the group
is finitely generated, nilpotent and torsion free. As a consequence we obtained
rationality and a functional equation for the Reidemeister zeta function of a
continuous map where the fundamental group of $X$ is as above.

In Chapter 3 we show that the Nielsen zeta function has a positive radius of convergence
which admits a sharp estimate in terms of the topological entropy of the map. We also
give an exact algebraic lower estimation for the radius.With the help of Nielsen - Thurston
theory of surface homeomorphisms we prove that for an orientation-preserving
homeomorphism of a compact surface the Nielsen zeta function is either a rational function
or the radical of rational function. For a periodic map of a compact polyhedron
we prove a product formula for Nielsen zeta function which implies that Nielsen zeta
function is a radical of a rational function.
In section 3.4 and 3.5 we give sufficient conditions under which the Nielsen zeta function
coincides with the Reidemeister zeta function and is a rational function with
functional equation.
In section 3.6 we describe connection between the rationality of the Nielsen zeta functions
for the maps of fiber, base and total space of a fiber map of a Serre bundle. We would like
to mention that in all known cases the Nielsen zeta function is a
nice function. By this we mean that it is a product of an exponential of a polynomial with
a function some power of which is rational. May be this is a general pattern.

In Chapter 4 we generalize the results of Chapter 2-3 to the Nielsen and Reidemeister zeta
functions modulo normal subgroup of the fundamental group.

In Chapter 5 we prove analog of Dold congruences for Reidemeister and Nielsen numbers.

In Chapter 6 we explain how dynamical zeta functions give rise to the Reidemeister torsion, a very
important topological invariant.
In section 6.2 we establish a connection between the Reidemeister torsion and Reidemeister zeta
function. We obtain an expression for the Reidemeister torsion of the mapping torus of the dual
map of a group endomorphism, in terms of the Reidemeister zeta function of the endomorphism.
This means that the Reidemeister torsion counts the fixed point classes of all iterates
of map $f$ i.e. periodic point classes of $f$.

In section 6.3 we establish a connection between the Reidemeister torsion of a mapping torus,
the eta-invariant, the Rochlin invariant and the multipliers of theta function.

In section 6.4 we describe with the help of the Reidemeister torsion and of an analog of Morse
inequalities the connection between the topology of the attraction domain of an attractor
and the dynamic of the system on the attractor.

In section 6.5 we show that for the integrable Hamiltonian system on the four-dimensional
symplectic manifold, the Reidemeister torsion of the isoenergetic surface counts the critical
circles(which are the closed trajectories of the system) of the second independent
Bott integral on this surface.

Keywords: dynamical zeta functions, Nielsen theory, Reidemeister and analytica l torsion, Nielsen and Reidemeister zeta functions