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We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeomorphic. Hairy Cantor sets appear in the study of the dynamics of holomorphic maps with infinitely many renormalisation structures. They are employed to link the fundamental concepts of polynomial-like renormalisation by Douady-Hubbard with the arithmetic conditions obtained by Herman-Yoccoz in the study of the dynamics of analytic circle diffeomorphisms.
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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.
It is established that any homeomorphism between two closed negligible subset of $D^tau$ can be extended to an autohomeomorphism of $D^tau$.
We define the epsilon-distortion complexity of a set as the shortest program, running on a universal Turing machine, which produces this set at the precision epsilon in the sense of Hausdorff distance. Then, we estimate the epsilon-distortion complexity of various central Cantor sets on the line generated by iterated function systems (IFSs). In particular, the epsilon-distortion complexity of a C^k Cantor set depends, in general, on k and on its box counting dimension, contrarily to Cantor sets generated by polynomial IFS or random affine Cantor sets.
The notion of Haar null set was introduced by J. P. R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke. During the last twenty years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. Inspired by these various results, we introduce the topological analogue of the notion of Haar null set. We call it Haar meager set. We prove some basic properties of this notion, state some open problems and suggest a possible line of investigation which may lead to the unification of these two notions in certain context.
In recent years much attention has been enjoyed by topological spaces which are dominated by second countable spaces. The origin of the concept dates back to the 1979 paper of Talagrand in which it was shown that for a compact space X, Cp(X) is dominated by P, the set of irrationals, if and only if Cp(X) is K-analytic. Cascales extended this result to spaces X which are angelic and finally in 2005 Tkachuk proved that the Talagrand result is true for all Tychnoff spaces X. In recent years, the notion of P-domination has enjoyed attention independent of Cp(X). In particular, Cascales, Orihuela and Tkachuk proved that a Dieudonne complete space is K-analytic if and only if it is dominated by P. A notion related to P-domination is that of strong P- domination. Christensen had earlier shown that a second countable space is strongly P-dominated if and only if it is completely metrizable. We show that a very small modification of the definition of P-domination characterizes Borel subsets of Polish spaces.
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