The local Hölder exponent for the dimension of invariant subsets of the circle

We consider for each (Formula presented.) the set (Formula presented.) of points of the circle whose forward orbit for the doubling map does not intersect (Formula presented.), and look at the dimension function (Formula presented.). We prove that at every bifurcation parameter (Formula presented.), the local Hölder exponent of the dimension function equals the value of the function (Formula presented.) itself. A similar statement holds for general expanding maps of the circle: namely, we consider the topological entropy of the map restricted to the survival set, and obtain bounds on its local Hölder exponent in terms of the value of the function.