Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion.

Intro -- Contents -- Introduction -- 0.1 From Riemann zeta function to dynamical zeta functions -- 0.1.1 Riemann zeta function -- 0.1.2 Problems concerning zeta functions -- 0.1.3 Important types of zeta functions -- 0.1.4 Hasse-Weil zeta function -- 0.1.5 Dynamical zeta functions -- 0.2 Dynamical zeta functions and Nielsen fixed point theory -- 0.3 Congruences for Reidemeister numbers -- 0.4 Reidemeister torsion -- 0.5 Table of contents -- 1 Nielsen Fixed Point Theory -- 1.1 History -- 1.2 Lifting classes and fixed point classes -- 1.2.1 The influence of a homotopy -- 1.3 Reidemeister numbers -- 1.3.1 Reidemeister numbers of a continuous map -- 1.3.2 Reidemeister numbers of a group endomorphism -- 1.4 Nielsen numbers of a continuous map -- 1.4.1 The fixed point index -- 1.4.2 Nielsen numbers -- 1.4.3 The least number of fixed points -- 2 The Reidemeister zeta function -- 2.1 A Convolution Product -- 2.2 Pontryagin Duality -- 2.3 Eventually commutative endomorphisms -- 2.3.1 Trace formula for the Reidemeister numbers of eventually commutative endomorphisms -- 2.3.2 Rationality of Reidemeister zeta functions of eventually commutative endomorphisms - first proof -- 2.3.3 Functional equation for the Reidemeister zeta function of an eventually commutative endomorphism -- 2.3.4 Rationality of Reidemeister zeta functions of eventually commutative endomorphisms - second proof -- 2.3.5 Connection of the Reidemeister zeta function with the Lefschetz zeta function of the dual map -- 2.4 Endomorphisms of finite groups -- 2.5 Endomorphisms of the direct sum of a free Abelian and a finite group -- 2.6 Endomorphisms of nilpotent groups -- 2.6.1 Functional equation -- 2.7 The Reidemeister zeta function and group extensions -- 2.8 The Reidemeister zeta function of a continuous map -- 2.8.1 The Reidemeister zeta function of a continuous map and Serre bundles.

3 The Nielsen zeta function -- 3.1 Radius of Convergence of the Nielsen zeta function -- 3.1.1 Topological entropy -- 3.1.2 Algebraic lower estimation for the Radius of Convergence -- 3.2 Nielsen zeta function of a periodic map -- 3.3 Orientation-preserving homeomorphisms of a compact surface -- 3.3.1 Geometry of the Mapping Torus and Radius of Convergence -- 3.4 The Jiang subgroup and the Nielsen zeta function -- 3.5 Polyhedra with finite fundamental group -- 3.6 Nielsen zeta function in other special cases -- 3.6.1 Pseudo-Anosov homeomorphism of a compact surface -- 3.7 The Nielsen zeta function and Serre bundles -- 3.8 Examples -- 4 Reidemeister and Nielsen zeta functions modulo normal subgroup, minimal dynamical zeta functions -- 4.1 Reidemeister and Nielsen zeta functions modulo a normal subgroup -- 4.1.1 Radius of Convergence of the mod K Nielsen zeta function -- 4.1.2 mod K Nielsen zeta function of a periodic map -- 4.2 Minimal dynamical zeta function -- 4.2.1 Radius of Convergence of the minimal zeta function -- 5 Congruences for Reidemeister and Nielsen numbers -- 5.1 Irreducible Representation and the Unitary Dual of G -- 5.2 Endomorphism of the Direct Sum of a Free Abelian and a Finite Group -- 5.3 Endomorphism of almost Abelian groups -- 5.3.1 Compact Groups -- 5.3.2 Almost Abelian groups -- 5.4 Endomorphisms of nilpotent groups -- 5.5 Main Theorem -- 5.6 Congruences for Reidemeister numbers of a continuous map -- 5.7 Congruences for Reidemeister numbers of equivariant group endomorphisms -- 5.8 Congruences for Reidemeister numbers of equivariant maps -- 5.9 Congruences for Nielsen numbers of a continuous map -- 5.10 Some conjectures for wider classes of groups -- 6 The Reidemeister torsion -- 6.1 Preliminaries -- 6.2 The Reidemeister zeta Function and the Reidemeister Torsion of the Mapping Torus of the dual map.

6.3 The connection between the Reidemeister torsion, eta-invariant, the Rochlin invariant and theta multipliers via the dynamical zeta functions -- 6.3.1 Rochlin invariant -- 6.3.2 The Bismut-Freed-Witten Holonomy Theorem -- 6.3.3 Connection with Reidemeister torsion -- 6.3.4 The Reidemeister torsion and theta-multipliers -- 6.4 Topology of an attraction domain and the Reidemeister torsion -- 6.4.1 Introduction -- 6.4.2 Morse-Smale systems -- 6.4.3 A formula for the Euler characteristic -- 6.4.4 The Reidemeister torsion of the level surface of a Lyapunov function and of the attraction domain of the attractor -- 6.5 Integrable Hamiltonian systems and the Reidemeister torsion.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.