The Principle of Open Induction on Cantor space and the Approximate-Fan Theorem

The paper is a contribution to intuitionistic reverse mathematics. We work in a weak formal system for intuitionistic analysis. The Principle of Open Induction on Cantor space is the statement that every open subset of Cantor space that is progressive with respect to the lexicographical ordering of Cantor space coincides with Cantor space. The Approximate-Fan Theorem is an extension of the Fan Theorem that follows from Brouwer's principle of induction on bars in Baire space and implies the Principle of Open Induction on Cantor space. The Principle of Open Induction in Cantor space implies the Fan Theorem, but, conversely the Fan Theorem does not prove the Principle of Open Induction on Cantor space. We list a number of equivalents of the Principle of Open Induction on Cantor space and also a number of equivalents of the Approximate-Fan Theorem

Cantor set zeros of one-dimensional Brownian motion minus Cantor function

It was shown by Antunovi\'{c}, Burdzy, Peres, and Ruscher that a Cantor function added to one-dimensional Brownian motion has zeros in the middle $\alpha$-Cantor set, $\alpha \in (0,1)$, with positive probability if and only if $\alpha \neq 1/2$. We give a refined picture by considering a generalized version of middle 1/2-Cantor sets. By allowing the middle 1/2 intervals to vary in size around the value 1/2 at each iteration step we will see that there is a big class of generalized Cantor functions such that if these are added to one-dimensional Brownian motion, there are no zeros lying in the corresponding Cantor set almost surely.Comment: 19 pages, improved Theorem 3.

Non-homeomorphic topological rank and expansiveness

Downarowicz and Maass (2008) have shown that every Cantor minimal homeomorphism with finite topological rank $K > 1$ is expansive. Bezuglyi, Kwiatkowski and Medynets (2009) extended the result to non-minimal cases. On the other hand, Gambaudo and Martens (2006) had expressed all Cantor minimal continuou surjections as the inverse limit of graph coverings. In this paper, we define a topological rank for every Cantor minimal continuous surjection, and show that every Cantor minimal continuous surjection of finite topological rank has the natural extension that is expansive

Diffusion on middle-ξ\xi Cantor sets

In this paper, we study $C^{\zeta}$-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the $C^{\zeta}$-calculus on the generalized Cantor sets known as middle-$\xi$ Cantor sets. We have suggested a calculus on the middle-$\xi$ Cantor sets for different values of $\xi$ with $0<\xi<1$. Differential equations on the middle-$\xi$ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.Comment: 15 pages, 11 figure