$K$-theory of endomorphisms, the $\mathit{TR}$-trace, and zeta functions

$K$-theoretic torsion and the zeta function

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.

Towards a dichotomy for the Reidemeister zeta function

We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups $\mathbb Z_p^d,$ where $\mathbb Z_p$ the group of p-adic integers. As a consequence, we obtain a dichotomy for the Reidemeister zeta function of a continuous map of a topological space with fundamental group that is non-finitely generated torsion abelian group. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphisms pairs of finitel…

同步点：增长、渐进行为、同余关系以及同步ζ函数

Synchronization points: growth, asymptotics, congruences, and the synchronization zeta function

本文研究了同步点的增长、渐进行为及同余关系，引入同步ζ函数并探讨其与拓扑熵和Reidemeister束缚的联系。

分级自同态的超迹缺陷

The Supertrace Defect of a Graded Endomorphism

本文受Igusa关系启发，为有界链复形上不交换微分的分级自同态构造泛商，迭代得极小滤过与平坦退化，使幂超迹由Lefschetz迹表示，给出对数行列式与zeta级数的同调表示。

Reidemeister 和 Nielsen 数：增长速率、渐进行为、动力学ζ函数和高斯同余

The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences

本文从动力学角度研究Reidemeister和Nielsen数序列的增长速率、渐进行为及高斯同余，证明了相关ζ函数的有理性和序列的生长特性。