Alec Norton

The Gauss-Green theorem for fractal boundaries

v is the 1-vectorfield “dual” to ω: if ω = ∑ (−1)fi dx1 ∧ · · · ∧ ∧ dxi ∧ · · · ∧ dxn, then v = (f1, . . . , fn).) There has been considerable effort in the literature (e.g. [JK], [M], [P]) to extend this formula to permit integrands of less regularity by generalizing the Lebesgue integral. On the other hand, invariably the situations in which (1) holds require fairly strong hypotheses on the boundary ∂Ω, e.g. that it should have sigmafinite (n−1)-measure, or that the gradient of the characteristic function of Ω be a vector valued measure with finite total variation [F], [P]. However there is a natural way to expand the validity of (1) to much more general boundaries while still using the ordinary Lebesgue integral; this is the topic of the present paper. For the case of Lipschitz forms, the results of this paper follow readily from Whitney’s theory of flat chains [W2]. However his approach to the Gauss-Green theorem is not widely appreciated because he focused on chains and cochains, where effectively (1) is used to define the exterior derivative. In [HN] we extend Whitney’s method to treat the more general Holder case. (Only the case n = 2 is discussed

Functions not constant on fractal quasi-arcs of critical points

This paper provides geometric sufficient conditions for an arc to be a critical set for some function not constant along that arc-an example of which was first discovered by Whitney in 1935. In particular, any fractal subarc of a quasi-circle has this property. The maximum degree of differentiability of the function is closely connected to the arcs geometry.

Denjoy's theorem with exponents

If X is the (unique) minimal set for a C1+α diffeomorphism of the circle without periodic orbits, 0 < α < 1, then the upper box dimension of X is at least α. The method of proof is to introduce the exponent α into the proof of Denjoy’s theorem.