tower construction will be required in Sections \[sec-pde\] and \[sec-L\].

Starting from the first return map $F=f^\varphi:Y\to Y$, there is a standard way of constructing an extension $f_\Delta:\Delta\to\Delta$ of the underlying map $f:X\to X$. Define the [*tower*]{} $\Delta=\{(y,j)\in Y\times{{\mathbb Z}}:0\le j<\varphi(y)\}$ and the [*tower map*]{} $f_\Delta:\Delta\to\Delta$ given by $f_\Delta(y,j)=(y,j+1)$ for $j\le \varphi(y)-2$ and $f_\Delta(y,j)=(Fy,0)$ for $j= \varphi(y)-1$.

The base of the tower $\{(y,0):y\in Y\}$ is naturally identified with $Y$ and so we may regard $Y$ as a subset of both $X$ and $\Delta$. Let $\mu_\Delta$ be the unique $f_\Delta$-invariant measure on $\Delta$ that agrees with the underlying measure $\mu$ on the common subset $Y$.

Define the projection $\pi:\Delta\to X$, $\pi(y,j)=f^jy$. Then $\pi f_\Delta = f \pi$ and $\pi_*\mu_\Delta=\mu$. Thus $f_\Delta$ is an extension of $f$ with the same first return map $F:Y\to Y$ and return time function $\varphi:Y\to{{\mathbb Z}}^+$ as the original map.

Asymptotics of $T(\theta)$ {#sec-T}
==========================

In this section, we obtain an asymptotic expression for $T(\theta)$ as $\theta\to0^+$. Throughout, $\beta\in(0,1)$. The main part of the analysis is to understand the asymptotics of the leading eigenvalue $\lambda(\theta)$. In certain situations, we obtain a higher order expansion. Let $c_\beta=-i\int_0^\infty e^{i\sigma}\sigma^{-\beta}\,d\sigma$.

\[lem-AD\] Let $\beta\in(0,1)$. As $\theta\to0^+$,

-   $\lambda(\theta)=1-c_\beta\ell(1/\theta)\theta^\beta(1+o(1))$.

-   $T(\theta)-(1-\lambda(\theta))^{-1}P=O(1)$.

-   