Critical Regularity of Invariant Foliations

:D AEDF .=G . The spectrum H $ I(KJEL of its complexification is called the Mather spectrum [M]. Anosov showed that and , called the unstable and stable bundles, are Hölder continuous. One an give lower bounds for attainable values of the Hölder exponent in terms of contraction and expansion rates of ! . Optimal estimates and reasons for wanting them are given in [H2, HW]. They are not exceeded for a generic symplectic Anosov diffeomorphism or flow, or the geodesic flow of a generic negatively curved metric. High regularity of the unstable subbundle is associated with rigidity. It is typically only possible in algebraic systems [HK, Gh, Kn, FK, BFL, BCG]. We illuminate a related aspect in the intermediate range of regularity. While it has been known for quite a while that moderately high finite differentiability implies infinite differentiability (if the diffeomorphism is M ) [HK, H1, FL], we describe circumstances where such regularity jumps occur below ,N . We forego the fullest generality to more clearly show the mechanism at work. The minimal level of Hölder regularity is guaranteed by standard relations between contraction and expansion rates. If the rings of the Mather spectrum are sufficiently narrow then one can apply the uniform version [HW] of the forward-matching method introduced in [H2] to show that this lowest regularity is exceeded outside a negligible set only if it is substantially higher everywhere (by [HW] the lowest regularity is generically not exceeded off a negligible set). There is also a jump to almost N if the second regularity level is substantially exceeded. I thank Northwestern University for inspiration and for their hospitality in January 2001, when the idea for this paper occurred to me. Definition 1.1. If O * we say that a function PRQ)ASCET is 3U or O -Hölder at VW5XA if there exist 8ZY []\ such that