Ergodic averages of commuting transformations with distinct degree polynomial iterates

We prove mean convergence, as $N\to \mathrm{\infty }$, for the multiple ergodic averages $\frac{1}{N}{\sum }_{n=1}^N{f}_1(T_1^{p_1(n)}x)\mathrm{.}\mathrm{.}\mathrm{.}{f}_{\mathrm{\ell }}(T_{\mathrm{\ell }}^{p_{\mathrm{\ell }}(n)}x)$, where $p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_{\mathrm{\ell }}$ are integer polynomials with distinct degrees, and $T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_{\mathrm{\ell }}$ are commuting, invertible measure preserving transformations, acting on the same probability space. This establishes several cases of a conjecture of Bergelson and Leibman, that complement the case of linear polynomials, recently established by Tao. Furthermore, we show that, unlike the case of linear polynomials, for polynomials of distinct degrees, the corresponding characteristic factors are mixtures of inverse limits of nilsystems. We use this particular structure, together with some equidistribution results on nilmanifolds, to give an application to multiple recurrence and a corresponding one to combinatorics.