  In the class of strictly convex smooth boundaries, each of which not having
strip around its boundary foliated by invariant curves, we prove that the
Taylor coefficients of the "normalized" Mather's $\beta$-function are
invariants under $C^\infty$-conjugacies. In contrast, we prove that any two
elliptic billiard maps are $C^0$-conjugated near their respective boundaries,
and $C^\infty$-conjugated in the open cylinder, near the boundary and away from
a plain passing through the center of the underlying ellipse. We also prove
that if the billiard maps corresponding to two ellipses are topologically
conjugated then the two ellipses are similar.
