In this paper we study the multiple ergodic averages
$$\frac{1}{n}\sum_{k=1}^n \varphi(x_k, x_{kq}, \cdots, x_{k q^{\ell-1}}),
\qquad (x_n) \in \Sigma_m$$
on the symbolic space $\Sigma_m =\{0, 1, \cdots, m-1\}^{\mathbb{N}^*}$ where $m\ge 2, \ell\ge 2, q\ge 2$ 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 measures) and the special case of telescopic measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying our multiplicatively invariant sets. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns with variational principle, pressure function and Legendre transform in this new setting.