Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation

Janvresse, Elise; de la Rue, Thierry; Velenik, Yvan

Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation

Authors

Janvresse, Elise

de la Rue, Thierry

Velenik, Yvan

Description

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l sum_{k=0}^{l-1} g(T^k x). When 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 $10,000 sequence.
version to appear in Stochastics and Dynamics. We added a discussion on the links with Conway 10,000$ recursive sequence