Mean ergodic theorems in $L^r(μ)$ and $H^r(\mathbb T)$, $0<r<1$
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| Format: | Preprint |
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2023
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| _version_ | 1866913181954211840 |
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| author | Abdalaoui, el Houcein el Lin, Michael |
| author_facet | Abdalaoui, el Houcein el Lin, Michael |
| contents | 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 $h \in {\mathcal F}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00567 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mean ergodic theorems in $L^r(μ)$ and $H^r(\mathbb T)$, $0<r<1$ Abdalaoui, el Houcein el Lin, Michael Dynamical Systems Functional Analysis 37A30, 42B30, 37E10 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 $h \in {\mathcal F}$. |
| title | Mean ergodic theorems in $L^r(μ)$ and $H^r(\mathbb T)$, $0<r<1$ |
| topic | Dynamical Systems Functional Analysis 37A30, 42B30, 37E10 |
| url | https://arxiv.org/abs/2401.00567 |