First Return Recovery of Baire Class One Functions on Ultrametric Spaces
In his 1992 paper First return path derivatives, R. J. O'Mally defines a First Return Path System, motivated by the Poincaré first return map of differential dynamics. While the original intent of the first return path system was to study questions of differentiability, the concept has found other applications. U. B. Darji and M. J. Evans applied first return paths to the question of recovering functions in Recovering Baire one Functions. The problem they posed was: for what types of functions f from X to Y, where X and Y are separable metric spaces, can we recover the function based only on a countable dense subset of the domain and a simple recursive procedure which produces a first return path. The conclusion they reached was that for compact metric spaces, the functions recoverable in this way are the Baire class one functions. The examination of this topic continued in How can we recover Baire class one functions? in which D. Lecomte showed that in certain ultrametric spaces, the class of recoverable functions are also exactly the class of Baire class one functions. Also in this paper, Lecomte gave an example of a Baire class one function on an ultrametric space which is not recoverable. I continue the study of first return recoverability on ultrametric spaces by examining the conditions required on an ultrametric space in order to either find a non-recoverable Baire class one function, or to show that all Baire class one functions are first return recoverable. I provide criteria for ultrametric spaces both for the recoverability of Baire class one functions and for the existence of non-recoverable Baire class one functions in terms of the distances obtained at individual points. This considerably generalize Lecomte's results and examples.
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