Example 14.19.41F8EB82A5AD2B

Example 14.19.41.F8EB82A5AD2B Choose $0<a<1/2$, and use the function $s=\log |z|^2$ on the covering space $\mathbb C^3\setminus \{ 0\} $ from section 23.1. Since $s$ changes by $-2$ under the defining contraction, every $2$-periodic function of $s$ descends to $X$. For an integer $m\geq 1$, define the smooth normalized potential

Primes below denote derivatives with respect to $s$. Since $\mathrm{dd}^{\mathrm{c}}s=\alpha $ and $\mathrm{d}s\wedge \mathrm d^{\mathrm c}s=\beta $, the chain rule gives

Both coefficients are bounded below by positive constants independent of $m$. The rank identities $\alpha ^3=\beta ^2=0$ imply

On each fiber, $s$ is twice the real Haar coordinate plus a constant depending on the base point, so its distribution modulo $2$ is uniform. When the periodic expression $H_ m=(1+u_ m')^2(1+u_ m'')$ is averaged, the terms $u_ m'$, $u_ m''$, $u_ m'u_ m''$ and $(u_ m')^2u_ m''$ contribute zero, since they are derivatives of periodic functions. Hence

Set $f_ m=H_ m/I_ m$. These smooth densities have unit integral and common strictly positive lower and finite upper bounds. Uniqueness of the constant gives $c_{f_ m}=I_ m\to 1$, and $Af_ m=1$. Nevertheless,

Here the constant in $O(m^{-1})$ depends only on $a$, and the last identity uses the fact that the average of $|\cos |$ over a period is $2/\pi $. Thus near equality does not imply strong $L^1$ convergence to the fiber average, even for uniformly positive bounded smooth densities. For any pullback $g$ of a function on the base, the contraction property gives $\| f_ m-Af_ m\| _1\leq 2\| f_ m-g\| _1$, so approximation by such pullbacks also fails in $L^1$.