Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics

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 ${\varphi }_{\beta }$ defined on $[0,1]$ (with a complex analytic extension) for the Ruelle operator of $\beta \log g$. Under some assumptions we show that $\frac{1}{\beta }\log {\varphi }_{\beta }$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when $\log g(x)=-\log f^{\mathrm{\prime }}(x)$. In that case we relate the involution kernel to the so called scaling function.