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 $β\,\log g$, where $β$ is a real constant, there exists a real analytic eigenfunction $ϕ_β$ defined on $[0,1]$ (with a complex analytic extension) for the Ruelle operator of $β\,\log g$. Under some assumptions we show that $\frac{1}β\, \log ϕ_β$ 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.

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