[Paper Review] On the leafwise cohomology and dynamical zeta functions for fiber bundles over the circle

Junhyeong Kim|arXiv (Cornell University)|Dec 12, 2017

Topological and Geometric Data Analysis3 references3 citations

TL;DR

This paper provides explicit formulas for leafwise cohomology and the regularized determinant of the dynamical zeta function for surface bundles over the circle, establishing a functional equation and special value formulas. It realizes Deninger's conjectural dynamical Lefschetz trace formula in a concrete geometric setting, linking transverse dynamics to cohomological invariants via monodromy action on fiber cohomology.

ABSTRACT

In this paper, we give concrete descriptions of leafwise cohomology groups and show the regularized determinant expression of the dynamical zeta function for fiber bundles over $S^{1}$. As applications, we show a functional equation and some formulas for special values of the dynamical zeta function.

Motivation & Objective

• To concretely describe leafwise cohomology groups for fiber bundles over $S^1$ equipped with a 1-codimensional foliation.
• To derive a regularized determinant expression for the dynamical zeta function in this geometric setting.
• To establish a functional equation for the dynamical zeta function using duality in cohomology.
• To compute special values of the zeta function using Lefschetz numbers and eigenvalue decomposition of the monodromy action.
• To provide a concrete realization of Deninger's conjectural framework for dynamical zeta functions in foliated geometry.

Proposed method

• Uses the suspension of a diffeomorphism $\varphi$ on a closed surface $S$ to construct a fiber bundle $M$ over $S^1$, inducing a 1-codimensional foliation $\mathcal{F}$.
• Expresses leafwise cohomology $H^i_{\mathcal{F}}(M)$ in terms of the singular cohomology $H^i(S, \mathbb{C})$ and the monodromy action $\varphi^*$.
• Applies the Lefschetz fixed-point theorem to relate periodic orbits of the $\mathbb{Z}$-action on $S$ to the dynamical zeta function $\zeta(M;s)$.
• Derives the regularized determinant expression $\zeta(M;s) = \prod_{i=0}^d \det(1 - \varphi^* r^{-s} \mid H^{d-i}(S,\mathbb{C}))^{(-1)^{i+1}}$.
• Uses Poincaré duality and the symmetry of the monodromy action to prove the functional equation $\zeta(M;s) = (-r^s)^{\chi(S)} \zeta(M;-s)$.
• Computes special values via eigenvalue decomposition of $\varphi^*$ and relates them to the Lefschetz numbers $\Lambda(\varphi^m)$.

Experimental results

Research questions

• RQ1 How can the leafwise cohomology groups of a surface bundle over $S^1$ be explicitly described in terms of fiber cohomology?
• RQ2 What is the regularized determinant expression for the dynamical zeta function in this setting?
• RQ3 Does the dynamical zeta function satisfy a functional equation, and if so, what is its symmetry?
• RQ4 How are special values of the zeta function related to the Lefschetz numbers of iterates of the monodromy?
• RQ5 Can Deninger's conjectural dynamical Lefschetz trace formula be realized concretely in this example?

Key findings

• The leafwise cohomology groups $H^i_{\mathcal{F}}(M)$ are isomorphic to $H^i(S, \mathbb{C})$ as vector spaces, with the infinitesimal generator $\Theta$ acting via the monodromy $\varphi^*$.
• The dynamical zeta function admits the regularized determinant expression $\zeta(M;s) = \prod_{i=0}^d \det(1 - \varphi^* r^{-s} \mid H^{d-i}(S,\mathbb{C}))^{(-1)^{i+1}}$.
• The zeta function satisfies the functional equation $\zeta(M;s) = (-r^s)^{\chi(S)} \zeta(M;-s)$, symmetric about $\mathrm{Re}(s) = 0$.
• The order of $\zeta(M;s)$ at $s = k$ is given by $\mathrm{ord}_{s=k}\zeta(M;s) = \sum_i (-1)^{i+1} \dim(H^i_{\mathcal{F}}(M)^{\Theta \sim k})$.
• The special value at $s = k$ is $\zeta(M;k)^* = (\log r)^{\mathrm{ord}_{s=k}\zeta(M;s)} \exp\left(\sum_{m \geq 1} \frac{r^{-km} \Lambda(\varphi^m) + \mathrm{ord}_{s=k}\zeta(M;s)}{m}\right)$.
• The Lefschetz number $\Lambda(\varphi^m) = \sum_i (-1)^i \mathrm{tr}(\varphi^{m*} \mid H^i(S))$ fully determines the logarithmic derivative of the zeta function at special points.