On Some Invariants of Birkhoff Billiards Under Conjugacy

In the class of strictly convex smooth boundaries each which has no strip around its boundary foliated by invariant curves, we prove that Taylor coefficients “normalized” Mather’s $$\beta$$ -function are under $$C^{\infty}$$ -conjugacies. contrast, any two elliptic billiard maps $$C^{0}$$ -conjugate near their respective boundaries, and -conjugate, away from a line passing through center underlying ellipse. We also that, if corresponding to ellipses topologically conjugate, then similar.