Papers:
32.
Differential complexes and exterior calculus, arxiv math-ph/0601015
31.
This gives a geometric dual to the Hodge star operator on differential forms. This plays the role of a normal vector field for smooth manifolds. It may be applied to a large class of domains, including some with infinite mass or tangents nowhere defined. Examples include the graph of the Weierstrass nowhere differentiable function.
30.
29.
Ravello Lecture Notes, Part I [pdf] , arXiv submission December 31, 2004, 83 pages. This was a rough draft that was posted, primarly as a date stamp. It is now replaced by an improved and expanded version liste above.
28.
27.
26.
This paper shows that a large subspace of currents can be geometrically realized as an infinite series of weighted simplexes, converging with respect to a norm defined in 25. The current applied to a form is the same as summing the integrals of the form over the weighted simplexes. This paper also shows that the generalized Stokes' theorem of 25 takes an optimal form . That is, all domains of integration for smooth forms satisfying minimal continuity conditions can be represented in this way.
25.
Papers 24 and 25 are fundamental to the theory of chainlets. The norms are defined in a coordinate free fashion. The isomorphism theorem implies both Stokes' and deRham's theorems and allows one to work at the level of chains and co-chains without passing to homology and cohomology.
24.
23.
This has the simplest exposition of the theory of chainlets in its early form. There are new developments that show how to express chainlets using an inner product structure.
22.
rth order conditionally convergent serie of fractal domains, Contemporary Mathematics, AMS, 203, (1997) 257--269.
21.
Numerical integration of vector fields over curves with zero area. Proceedings of the American Math Society. Vol 121, No. 3, July, 1994.
20.
Stokes' theorem for nonsmooth chains. Bulletin of the American Math Society, October, 1993.
19.
18.
A remark on the loxodromic mapping conjecture (with Charles Pugh). Bulletin of the Australian Math Society, Vol. 45, No. 3, pp. 521-524, 1992.
17.
Geometric integration on fractal curves in the plane (with Alec Norton). Indiana Journal, Vol. 40, pp. 567-594, 1991.
16.
The loxodromic mapping problem. Journal of Differential Equations, Vol. 90, No. 1, pp. 136-142, 1991.
15.
Geometry of algebraic continued fractals. London Math Society, Lecture Notes Series 134, Number Theory and Dynamical Systems, pp. 117-136, 1989.
14.
Embedded continued fractals and their Hausdorff dimension. Constructive Approximation, Springer Verlag, vol. 5, pp 99-115, 1989.
13.
An introduction to fractals. American Math Society, Proceedings of Symposia in Applied Mathematics vol. 39, pp. 107-126, 1989.
12.
A fixed point free ergodic flow on the three sphere (with Charles Pugh). Michigan Journal of Math, vol. 36, no. 2, pp. 261-266, 1989.
11.
Dynamics on Ahlfors quasi-circles. Proceedings of the Indian Academy of Sciences. (Math.Sci.) vol. 99, no. 2, pp. 113-122, 1989.
10.
Denjoy fractals. Topology, vol. 28, no. 1, pp. 59-80, 1989.
9.
C2 counterexamples to the Seifert conjecture. Topology, vol. 27, no. 3, pp. 249-278, 1988.
8.
Continued fractals and the Seifert Conjecture. Bulletin of the American Math Society, vol. 13, no. 2, pp. 147-153, 1985.
7.
Flows on S3 and R3 without periodic orbits (With James A. Yorke). Lecture Notes in Mathematics 1007, (refereed) pp. 401-407, 1983.
6.
Opening closed leaves of foliations. Bulletin of the London Math Society, vol. 15, pp. 218-220, 1983.
5.
Wandering intervals. Lecture Notes in Mathematics 898, (refereed) Springer Verlag, pp.154-163, 1981.
4.
Unsmoothable diffeomorphisms on higher dimensional manifolds. Proceedings of the American Math Society, vol. 73, pp.249-255, 1979.
3.
Structure of a foliated neighborhood. Proceedings of the Cambridge Philosophical Society (Cambridge, England), vol. 79, pp.101-110, 1976.
2.
Unsmoothable diffeomorphisms. Annals of Mathematics, vol. 102, pp. 85-94, 1975.
1.
On unsmoothable diffeomorphisms. Bulletin of the American Math Society, vol. 81, p. 746, 1975.
Special Books and Articles:
3.
Chainlet Geometry. (in preparation) to be published by the Advanced Book Program of Perseus Books (formerly Addison-Wesley).
This text will detail Harrison's theory of chainlets, a theory dual to those of wavelets and Fourier Series. The text will include optimal extensions of the theorems of Stokes', Green, and Gauss with applications to algebraic topology, differential topology, fractal geometry, Lebesgue integration and measure, numerical analysis, and physics.
2.
The Escher Staircase. Notices of the American Math Society, September, 1991.
1.
Continued Fractals. review by Ian Stewart, Nature, December 1985.