MAPPING TORUS AND THE ASYMPTOTIC EXPANSION OF ${\log}\;T(M,{\varphi})(t)$

In this paper we define a torsion function ${\log}\;T(M,{\varphi})(t)$) for $t{\gg}0$ and show that it has an asymptotic expansion $\frac{1}{2}{\chi}(M)t$ as $t{\rightarrow}{\infty}$.