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

We consider for each t the set K(t) of points of the circle whose forward orbit for the doubling map does not intersect (0,t), and look at the dimension function eta(t) := H.dim K(t). We prove that at every bifurcation parameter t, the local Hoelder exponent of the dimension function equals the value of the function eta(t) itself. The same statement holds by replacing the doubling map with the map g(x) := dx mod 1 for d >2.