The rapid points of a complex oscillation
Paul Potgieter (Unisa)
TL;DR
This paper investigates the Hausdorff dimension of rapid points in Brownian motion and extends the results to complex oscillations, demonstrating they share the same dimensional properties.
Contribution
It proves that complex oscillations have the same Hausdorff dimension of rapid points as Brownian motion, using a counting argument on sample paths.
Findings
Rapid points of Brownian motion have a specific Hausdorff dimension.
Complex oscillations share the same rapid point dimension as Brownian motion.
The proof technique can be applied to algorithmically random Brownian motion.
Abstract
By considering a counting-type argument on Brownian sample paths, we prove a result similar to that of Orey and Taylor on the exact Hausdorff dimension of the rapid points of Brownian motion. Because of the nature of the proof we can then apply the concepts to so-called complex oscillations (or 'algorithmically random Brownian motion'), showing that their rapid points have the same dimension.
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