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Poincaré rotation numbers and Riesz and Voronoi means. (English. Russian original) Zbl 1073.37049
Math. Notes 74, No. 2, 299-301 (2003); translation from Mat. Zametki 74, No. 2, 314-315 (2003).
We say that a sequence of numbers (not necessarily converging) converges in Cesàro sense, if their averages converge. We say that it converges in the sense of Riesz with given weights \(p_0, p_1,\dots\), \(p_0>0\), \(p_i\geq0\), if their averages with the weights \(p_i\) converge. We say that a sequence converges in the sense of Voronoi with given weights as above, if the latter convergence statement holds true, but for averages taken with the inverse order of weights. Given a circle homeomorphism \(T:S^1\to S^1= \mathbb R/2\pi\mathbb Z\), \(T(x)=x+f(x)\). Poincaré’s theorem implies that for any \(x\) the sequence \(T^n(x)\) converges to the rotation number of \(T\) in the sense of Cesàro. The paper under review gives sufficient conditions on a weight sequence \(p_i\) for convergence of a sequence \(T^n(x)\) (with arbitrary \(T\) and \(x\)) in either Riesz or Voronoi sense. The first result (Theorem 1) affirms Voronoi convergence of \(T^n(x)\) , whenever the weight sequence \(p_i\) is nonincreasing and forms a diverging series. The second result (Theorem 2) affirms Riesz convergence of \(T^n(x)\), whenever the weights \(p_i\) are nondecreasing and in addition \(\frac{p_n} {\sum_0^np_s}\to0\). In fact, Theorem 1 and 2 are equivalent. The authors also give the following weaker sufficient condition for Riesz convergence in Theorem 2: \(\frac{p_n} {\sum_0^np_s}\to0\), and there exists an \(H>0\) such that for any \(n\) one has \[ p_0(n+1)+| p_1-p_0| n+\dots+ | p_n-p_{n-1}| \leq H(p_0+\dots+p_n). \] Theorem 2 follows from Theorem 3. In the proof of Theorem 3 the authors use results from [G. H. Hardy, Divergent series, Oxford: At the Clarendon Press (Geoffrey Cumberlege) XIV (1949; Zbl 0032.05801)] on relation between Voronoi and Cesàro convergences.

37E45 Rotation numbers and vectors
37E10 Dynamical systems involving maps of the circle
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