Distribution of polynomial orbits in Toeplitz systems

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study the density of the polynomial orbits in Toeplitz systems—we show that it implies equidistribution of the polynomial orbits in the class of regular Toeplitz systems, but not in the class of strictly ergodic ones.