Large deviations for multiple ergodic averages

VERICAT, FERNANDO; MESON, ALEJANDRO M.

JOURNAL OF INTERDISCIPLINARY MATHEMATICS

Taylor & Francis

Lugar: NEW DELHI; Año: 2017 vol. 20 p. 1603 - 1617

0972-0502

The main purpose of this work is to estimate how multiple ergodic averages appart from a given quantity. This problem can be studied by describing a large deviation process for empirical measures as obtained by using the contraction principle. The case of single ergodic averages for empirical measures was already studied by Pfister and Sullivan [Nonlinarity, 10 (2005) 237-261]. To have a more complete picture on empirical measures and V-statistics, we estimate the size of the sets G_{K}={x:L_{r}(x)⊂K}, where L_{r}(x) is the limit-point set of the sequence of empirical measures and K is a compact subset of M(X^{r}) with M(X) the set of measures on X. In pasrticular, we obtain a variational formula for the topological entropy of G_{K}. The result of this work about the dimension of the sets G_{K} can be compared with the one recently circulated by Fan, Schemeling and Wu [arXiv:1206.3214v1 (2012)].