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

TL;DR

This paper investigates the local Hölder exponent of the dimension function for invariant sets of the doubling map and similar maps, establishing a precise relationship between the exponent and the dimension at bifurcation points.

Contribution

It proves that the local Hölder exponent of the dimension function equals the function value at bifurcation points for the doubling map and maps with degree greater than two.

Findings

The local Hölder exponent equals the dimension function value at bifurcation points.

The result extends to maps of the form g(x) = dx mod 1 for d > 2.

The dimension function's regularity is characterized at bifurcation parameters.

Abstract

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.