Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theoremWhen seen as graphs of functions defined on {0, . . . , ℓ − 1}, we show for a suitable class of functions g that these quantities, once properly renormalized, converge to (part of) the graph of a self-affine function. The latter only depends on the ergodic component of x, and is a deformation of the so-called Blancmange function. We also briefly describe the links with a series of works on Conway recursive $10,000 sequence.
Abstract. The Poisson entropy of an infinite-measure-preserving transformation is defined in the 2005 thesis of Roy as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel's and Parry's entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy, − q i p i,j log p i,j , holds for any definitions of entropy. Poisson entropy dominates Parry's entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel's entropies equals difference of the Poisson entropies. In case there already exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised by Aaronson and Park about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations.
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