On the regularity of de~Rham curves

V. Yu. Protasov

Izvestiya. Mathematics , Tome 68 (2004) no. 3, pp. 567-606

Résumé

De Rham curves are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio $\omega:(1-2\omega):\omega$, where $\omega\in(0,1/2)$ is a given parameter. We find explicitly the sharp exponent of regularity of such a curve for any $\omega$. Regularity is understood in the natural parametrization using the arclength as a parameter. We also obtain a formula for the local regularity of a de Rham curve at each point and describe the sets of points with given local regularity. In particular, we characterize the sets of points with the largest and the smallest local regularity. The average regularity, which is attained almost everywhere in the Lebesgue measure, is computed in terms of the Lyapunov exponent of certain linear operators. We obtain an integral formula for the average regularity and derive upper and lower bounds.