Ergodic Theorems for Homogeneous Dilations

TL;DR

This paper establishes a general ergodic theorem for R^d actions, focusing on homogeneous dilations of Rajchman measures, with results on mean and pointwise convergence applicable to smooth submanifolds and polynomial curves.

Contribution

It introduces a new ergodic theorem for homogeneous dilations of Rajchman measures, extending classical results to broader settings with convergence criteria.

Findings

Proves mean convergence for Rajchman measures in Hilbert spaces.

Provides a Fourier dimension criterion for pointwise convergence.

Applies results to averages over submanifolds and polynomial curves.

Abstract

In this paper we prove a general ergodic theorem for ergodic and measure preserving actions of R^d on standard Borel spaces. In particular, we cover R.L. Jones ergodic theorem on spheres. Our main theorem is concerned with ergodic averages with respect to homogeneous dilations of Rajchman measures on Rd . We establish mean convergence in Hilbert spaces for general Rajchman measures, and give a criterion in terms of the Fourier dimension of the measure when almost everywhere pointwise convergence holds. Applications include averages over smooth submanifolds and polynomial curves.