## Some results on nonlinear recurrence.(English)Zbl 0803.28011

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.

### MSC:

 28D10 One-parameter continuous families of measure-preserving transformations 28D05 Measure-preserving transformations 37A99 Ergodic theory
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### References:

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