Mean ergodic theorems in $L^r(μ)$ and $H^r(\mathbb T)$, $0<r<1$
el Houcein el Abdalaoui, Michael Lin
TL;DR
This work analyzes mean ergodic behavior of the Koopman operator T=f∘θ on L^r(μ) for 0<r<1 and on Hardy spaces H^r(𝕋). It proves that if M_n f converges in L^r, the limit is T-invariant, hence constant under ergodicity, and it shows that M_n f→0 iff f lies in the closure of (I−T)L^r, though the converse fails in general; there exist functions in this closure for which M_n f does not converge a.e. Additionally, for p<1/r there exist dense G_δ subsets of L^p with limsup behavior of T^n h scaled by n^r diverging a.e. In the Hardy setting with irrational circle rotations, H^r(𝕋) admits a decomposition into constants and the closure of (I−T)H^r(𝕋), and M_N f converges in L^r to a(f) for each f∈H^r, while 1∉cl(I−T)H^r; nonetheless, there exist f∈(I−T)H^r with no a.e. convergence. An appendix provides delicate rate-of-convergence constructions showing sharp, nonuniform rates and limsup phenomena for T^n h.
Abstract
Let $T$ be the Koopman operator of a measure preserving transformation $θ$ of a probability space $(X,Σ,μ)$. We study the convergence properties of the averages $M_nf:=\frac1n\sum_{k=0}^{n-1}T^kf$ when $f \in L^r(μ)$, $0<r<1$. We prove that if $\int |M_nf|^r dμ\to 0$, then $f \in \overline{(I-T)L^r}$, and show that the converse fails whenever $θ$ is ergodic aperiodic. When $θ$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\int |M_nf|^r dμ\to 0$). We further establish that for $1 \leq p <\frac{1}{r},$ there is a dense $G_δ$ subset ${\mathcal F}\subset L^p(X,μ)$ such that $\limsup_n \frac{|T^nh|}{n^r}=\infty$ a.e. for any $h \in {\mathcal F}$.