The goal of the first part of this note is to get an explicit formula for rotation number and Mather β-function for ellipse. This is done here with the help of non-standard generating function of billiard problem. In this way the derivation is especially simple. In the second part we discuss application of Mather β-function to rigidity problem.
Keywords: Birkhoff billiard; action minimizers: rigidity, integrable billiards; invariant curve.