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Some sufficient conditions for the ergodicity of the Lévy transformation

We propose a possible way of attacking the question posed originally by Daniel Revuz and Marc Yor in their book published in 1991. They were asking whether the Lévy transformation of the Wiener--space is ergodic. Our main results are formulated in terms of a strongly stationary sequence of random variables obtained by evaluating the iterated paths at time one. Roughly speaking, this sequence has to approach zero "sufficiently fast". For example, one of our results states that if the expected hitting time of small neighbourhoods of the origin do not growth faster then the inverse of the size of these sets then the Lévy transformation is strongly mixing, hence ergodic.