Abstract. We first give a geometric characterization of ω-limit sets. We then use this characterization to prove that the family of ω-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.
It has recently been established that any Baire class one function f : [0, 1] → R can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of f . Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0, 1], the first coordinates of the vertices of the polygonal functions can be chosen from D.
We present a new characterization of Lebesgue measurable functions; namely, a function f : [0, 1] → R is measurable if and only if it is first-return recoverable almost everywhere. This result is established by demonstrating a connection between almost everywhere first-return recovery and a first-return process for yielding the integral of a measurable function.
We consider real-valued functions defined on 0, 1 . Both the class of Baire one functions and the class of Baire-one, Darboux functions have been characterized using first return limit notions. The former class consists of the first return recoverable functions and the latter consists of the first return continuous functions. Here we introduce a natural intermediate type of first return limiting process, first return approachability, and show that the first return approachable functions are precisely those Baire class one functions whose graphs are dense in themselves. Also, the set of points at which a function is first return approachable, but not first return continuous, is shown to be -porous. ᮊ
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