Multifractal analysis of some multiple ergodic averages

In this paper we study the multiple ergodic averages 1/n Sigma(n)(k=1) phi(x(k), x(kq), . . . , x(kq)(l-1)), (x(n)) is an element of Sigma(m) on the symbolic space Sigma(m) = {0,1, . . . , m -1}N*. where m >= 2, l >= 2, q >= 2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large plass of measures (called telescopic product measures). The special case of telescopic product measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying the level sets of the limit, which are not shift-invariant. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns variational principle, pressure function and Legendre transform in this new setting. (C) 2016 Elsevier Inc. All rights reserved.