Some results on nonlinear recurrence
The authors are interested in continuous time analogies of the nice properties that discrete measure-preserving systems exhibit along polynomial sequences. (For example, if \(p(t)\) is a polynomial with integer coefficients and \(f\in L^ 2(X,{\mathcal B},\mu)\) then the averages \({1\over N} \sum^ N_{n=1} f(T^{p(n)} x)\) converge in \(L^ 2\)-norm, where \(T\) is a measure-preserving transformation.) The situation turns out to be quite different for \(\mathbb{R}\) actions. Let \(\{S^ t\}_{t\in \mathbb{R}}\) be a continuous one-parameter flow of measure-preserving transformations of a Lebesgue probability space \((X,{\mathcal B},\mu)\). A sequence of real numbers \(\{\lambda_ n\}\) is said to be \(L^ p(T)\)-ergodic \((\{\lambda_ n\}\in E_ p(T))\), \(1\leq p\leq \infty\), if for every \(f\in L^ p(T)\), \(T= \mathbb{R}/\mathbb{Z}\), the sequence \(A_ N(f)= {1\over N} \sum^ N_{n=1} f(x+ \lambda_ n)\) is convergent a.e. (where \(x+ \lambda_ n\) is taken modulo 1). Theorem A. Let \(\lambda= \{\lambda_ n\}\) be a sequence of real numbers independent over \(\mathbb{Q}\). Then \(\lambda\not\in E_ \infty(T)\). Theorem B. For any \(\alpha\in \mathbb{Q}/\mathbb{Z}\), we have \(n^ \alpha= \{n^ \alpha\}\not\in E_ \infty(T)\). Theorem C. Let \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ k\in (0,1)\), \(\alpha_ i\neq \alpha_ j\) for \(i\neq j\). Let \(\{S^ t\}\) be an ergodic measure-preserving continuous flow acting on \((X,{\mathcal B},\mu\}\). Then for any \(f_ 1,f_ 2,\dots,f_ k\in L^ \infty(X,{\mathcal B},\mu)\) one has \[ \lim_{N\to \infty} \left\|{1\over N} \sum^ N_{n=1} \prod^ k_{i=1} f_ i(S^{n^{\alpha_ i}}x)- \prod^ k_{i=1} \int_ X f_ i d\mu\right\|_{L^ 2}= 0. \] The authors also obtain interesting consequences of these results and results on densities paralleling the discrete case.
ergodic measure-preserving continuous flow
scientific article; zbMATH DE number 569428