Uncountable absorbing systems in a functional space related to the Hausdorff dimension
by
Natalia Mazurenko
Precarpathian National University

We are interested in description of topology of some uncountable system of sets of continuous functions on the n-dimensional cube Iⁿ, where for these functions the Hausdorff dimension of their graphs takes the fixed values from the certain ordered set (from obvious reasons it follows that such dimension can take the values from an interval [n, n+1)).

If A is a countable infinite set then we denote s(A)=(-1, 1)^A (observe that s(A) ≅ s, where s=(-1, 1)^w is the pseudo-interior of the Hilbert cube Q=[-1, 1]^w) and define the following subspace of s(A): S(A) consists of all x=(x_a)_{a ∈ A} in s(A) such that sup_{a ∈ A}|x_a| < 1. It is clear that S(A) is a s-compact subset of s(A). If A=N then S(N)=S is the radial interior of the Hilbert cube Q.

For n ∈ N and any g ∈ [n, n+1) we denote C _{> g}(Iⁿ)={f ∈ C(Iⁿ) | dim_H(graphf) > g}. Here C(Iⁿ) is the space of continuous functions on Iⁿ and dim_H stands for the Hausdorff dimension.

As usual, by Q we denote set of rational numbers. The aim of the talk will be that the system {C _{> g}(Iⁿ)}_{g ∈ [n, n+1)} is a decreasing F_s-absorbing system in C(Iⁿ) (in sense of [1]) and there exists a homeomorphism a from C(Iⁿ) onto s([n, n+1)∩Q) such that for every g ∈ (n, n+1)

a(C > g(In))=S([n, g]∩Q)×s((g, n+1)∩Q).

This result extends that from [4] where it is proved that for any sequence (g_i), n ≤ g₁ < g₂ < ... < n+1, the sequence of the sets of functions in C(Iⁿ) whose graphs are of Hausdorff dimension > g_i forms an F_s-absorbing sequence in C(Iⁿ).

References:

[1] Cauty R. Strong Universality and Its Applications // Proc. of the Steklov Institute of Math. 212 (1996), 89-114.

[2] Dijkstra Jan J., Mogilski J. The topological product structure of systems of Lebesgue spaces // Math. Ann. 290 (1991), 527-543.

[3] Gladdines H. Absorbing systems in infinite-dimensional manifolds and applications. - Amsterdam: Vrije Universiteit, 1994. - 117 p.

[4] Mazurenko N., Absorbing systems in a function space, which are connected with the Hausdorff dimension // Matem. Studii 23:2 (2005), 207-216.

Date received: May 8, 2008