Equality of Hölder exponents for distribution functions of Gibbs measures

Abstract Pointwise Hölder exponents describe the degree of regularity of a function near a point. For a function f : R → R , a number α > 0 and a point t 0 ∈ R , write f ∈ C α ( t 0 ) if there exist a constant C > 0 , a number h > 0 and a polynomial P of degree less than α such that | f ( t ) − P ( t − t 0 ) | ⩽ C | t − t 0 | α for all t ∈ ( t 0 − h , t 0 + h ) . The pointwise Hölder exponent of f at t 0