Differentiability of fractal curves

Advances in Mathematics

This paper studies a large class of continuous functions f : [0, 1] → R d whose range is the attractor of an iterated function system {S 1 ,. .. , S m } consisting of similitudes. This class includes such classical examples as Pólya's space-filling curves, the Riesz-Nagy singular functions and Okamoto's functions. The differentiability of f is completely classified in terms of the contraction ratios of the maps S 1 ,. .. , S m. Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) f is nowhere differentiable; (ii) f is non-differentiable almost everywhere but with uncountably many exceptions; or (iii) f is differentiable almost everywhere but with uncountably many exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and (iii) above is calculated, and more generally, the complete multifractal spectrum of f is determined.

2004

The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as 'similarity' maps. Selfsimilar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of 'similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for self-similar fractals. We also prove a fractal version of Roth's theorem for algebraic points on a variety approximated by elements of a fractal subset. As a consequence, we get a fractal version of Siegel's theorem on finiteness of integral points on hyperbolic curves and a fractal version of Falting's theorem on Diophantine approximation on abelian varieties.

Proceedings of the American Mathematical Society, 1992

For self-similar sets with nonoverlapping pieces, Hausdorff dimension and measure are easily determined. We express "absence of overlap" in terms of discontinuous action of a family of similitudes, thus improving the usual "open set condition".

Nonlinearity, 2018

Proceedings of the American Mathematical Society, 1992

For self-similar sets with nonoverlapping pieces, Hausdorff dimension and measure are easily determined. We express "absence of overlap" in terms of discontinuous action of a family of similitudes, thus improving the usual "open set condition".

Physical Review A, 1991

... Although this approach has been very successful, up to now the related ideas (multifractality ... shown in (b)]. De-pending on the parameters b, different structures with arbitrary ... proper-ties can be investigated by the corresponding methods developed for such geometrical objects. ...

Fractals, 2011

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The Fα-integral and Fα-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and Fα-differentiability is generalized. Finally we touch upon an example of ab...

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Fractals, 2009

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called Fα-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called Fα-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The Fα-derivative is local unlike the classical fractional derivative. The Fα-calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved. The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension. Spaces of Fα-differentiable and Fα-integrable functions are analyzed. Analogues of Sobolev Spaces are construct...

Contemporary Mathematics, 2013

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.