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

Abstract

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.

The theory of filtrations of subalgebras, standardness, and independence

“…6 The theory of adic transformations and adic realizations of automorhisms became a new source of examples in ergodic theory. Even the first example suggested by the author, that of the Pascal automorphism, still intrigues researchers; it is not yet known whether its spectrum is continuous, and many other properties are also unknown (see [75], [79], [42], [24]).…”

The pascal automorphism has a continuous spectrum

“…This problem for the Pascal automorphism, along with its definition, was suggested by the author [6] in 1980 and subsequently considered in a series of papers (e.g., [14,16,18,17,19]), where various useful properties of the Pascal automorphism were studied; however, the problem has not been solved up to now. …”

Автоморфизм Паскаля Имеет Непрерывный Спектр

“…Эта задача для автоморфизма Паскаля вместе с его опреде-лением была поставлена автором в 1980 г. [6] и рассматривалась позже в серии работ [14], [16]- [19] и др, где были изучены различные полезные свойства авто-морфизма Паскаля, однако задача до сих пор не была решена.…”