Fractal analysis for sets of non-differentiability of Minkowski's question mark function

Abstract

In this paper we study various fractal geometric aspects of the Minkowski question mark function Q. We show that the unit interval can be written as the union of the three sets Λ 0 : = { x : Q ′ ( x ) = 0 } , Λ ∞ : = { x : Q ′ ( x ) = ∞ } , and Λ ∼ : = { x : Q ′ ( x ) does not exist and Q ′ ( x ) ≠ ∞ } . The main result is that the Hausdorff dimensions of these sets are related in the following way: dim H ( ν F ) < dim H ( Λ ∼ ) = dim H ( Λ ∞ ) = dim H ( L ( h top ) ) < dim H ( Λ 0 ) = 1 . Here, L ( h top ) refers to the level set of the Stern–Brocot multifractal decomposition at the topological entropy h top = log 2 of the Farey map F, and dim H ( ν F ) denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with F. The proofs rely partially on the multifractal formalism for Stern–Brocot intervals and give non-trivial applications of this formalism.