  We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of
degree $d$ which preserves orientation, and a real analytic positive potential
$g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex
analytic extension to a neighborhood of the interval in the complex plane. We
also assume $\log g$ is well defined for this extension.
  It is known in Complex Dynamics that under the above hypothesis, for the
given potential $\beta \,\log g$, where $\beta$ is a real constant, there
exists a real analytic eigenfunction $\phi_\beta$ defined on $[0,1]$ (with a
complex analytic extension) for the Ruelle operator of $\beta \,\log g$.
  Under some assumptions we show that $\frac{1}{\beta}\, \log \phi_\beta$
converges and is a piecewise analytic calibrated subaction. Our theory can be
applied when $\log g(x)=-\log f'(x)$. In that case we relate the involution
kernel to the so called scaling function.
