Joint ergodicity of Hardy field sequences

Abstract:

We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, t\log t$ and $e^{\sqrt {\log t}}$. We show that if all non-trivial linear combinations of the functions $a_1$, …, $a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $\frac {1}{N} \sum _{n=1}^{N}f_1(T^{\lfloor {a_1(n)}\rfloor }x)\cdot \dots \cdot f_k(T^{\lfloor {a_k(n)}\rfloor }x)$ exists and is equal to the product of the integrals of the functions $f_1$, …, $f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1$, …, $a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.