Hausdorff Dimension and Derivatives of Typical Nondecreasing Continuous Functions

Dimension de Hausdorff et dérivées des fonctions croissantes typiques

Alain Riviere (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

We endow the space Cr of nondecreasing functions on the unit interval [0, 1] with the uniform metric and consider its subspace Ccr of continuous nondecreasing functions. Then, we mainly prove: 1) In the sense of Baire categories, for most f ∈ Ccr, and also for most f ∈ Cr, we have: (a) the Stieljes measure mes f of f is carried by a set of Hausdorff dimension zero, (b) More precisely, f has zero left and right Diny lower derivatives everywhere outside a set of Hausdorff dimension zero, (c) For any 0 ≤ α ≤ ∞, the set of all t ∈ [0, 1] at which α is the left and right Diny upper derivative of f , is of Hausdorff dimension 1. 2) For most f ∈ Ccr, we have: (a) the set of all t at which f has an infinite Diny derivative contains a Cantor set in any nonempty open subset of [0, 1], (b) the same is true for the set of t at which f has positive and finite Diny lower derivative. 3) If A is any countable subset of [0, 1], then for most f ∈ Cr (a) mes f is atomic, and f is (left and right) discontinuous at each t ∈ A, (b) f has a zero lower derivative at each of its continuity point. Some other properties are proved which often mean that the fact that the measure mes f is carried by a set of null Hausdorff dimension does not imply similar properties for the derivative of f. 1 We give direct and elementary proofs of all these properties, except (1c)(when 0 < α < ∞) which is the more tricky one, and except (2b) and (3b) for the proof of which we need a geometric approach, using closely results and methods of [9]. Finally, we also explain how the property 1 c can be adapted to the case of typical functions with bounded variations.

Mots clés: Typical monotone function; Diny derivatives; upper Hausdorff dimension of a measure; convex function; curvature; cut locus

Domaines: Mathématiques [math]

Alain Riviere. Hausdorff Dimension and Derivatives of Typical Nondecreasing Continuous Functions. 2014. ⟨hal-01154558⟩