On the existence of functions with perfect level sets.
Investigation of smooth functions and analytic sets using fractal dimensions
We start from the following problem: given a function what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of functions. We investigate the analogous problem for functions. These are in a certain way intermediate between and functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.
Prolungamenti simultanei di misure
Nonatomic states
Chaotic behaviour of the map x ↦ ω(x, f)
Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and...
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