Abstract

Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation É. Janvresse, T. de la Rue and Y. Velenik Stoch. Dyn. 5, No 1, 1-25 (2005). Let $T$ be the Pascal-adic transformation. For any measurable function $g$, we consider the corrections to the ergodic theorem
$\sum_{k=0}^{j-1} g(T^k x) - \frac jL \sum_{k=0}^{L-1} g(T^k x).$
When seen as graphs of functions defined on $\{0,\ldots,L-1\}$, 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.
Key words: Pascal-adic transformation, ergodic theorem, self-affine function, Blancmange function, Conway recursive sequence.