Multifractal analysis of some multiple ergodic averages

In this paper we study the multiple ergodic averages1n∑k=1nφ(xk,xkq,⋯,xkqℓ−1),(xn)∈Σm on the symbolic space Σm={0,1,⋯,m−1}N⁎Σm={0,1,⋯,m−1}N⁎ where m≥2,ℓ≥2,q≥2m≥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 class 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.