Properties of Wiener-Wintner dynamical systems (Q2773556)

Let \((X,{\mathcal B},\mu,T)\) be a dynamical system. A function \(f\) is said to be a Wiener-Wintner of power type \(\beta\) in \(L^1\) if there exists a positive constant \(C_f\) and \(\beta\) such that NEWLINE\[NEWLINE\Biggl\|\sup_\varepsilon \Bigl|{1\over N}\sum^N_{n=1} f(T^nx)e^{2\pi in\varepsilon} \Bigr|\Biggr\|_1 \leq{C_f \over N^\beta}NEWLINE\]NEWLINEfor all positive integers \(N\). The system \((X,{\mathcal B},\mu,T)\) is said to be Wiener-Wintner dynamical system of power type \(\beta\), if there exists a dense set (in \(L^2)\) of Wiener-Wintner functions in \({\mathcal K}^\perp\), the orthocomplement of the Kronecker factor of \(T\). In a previous work of the first author [C. R. Acad. Sci., Paris, Sér. I, Math. 332, 321-324 (2001; Zbl 0983.37006)], several well-known dynamical systems were shown to be Wiener-Wintner but all the examples given had Lebesgue spectrum in \({\mathcal K}^\perp\). In the article under review, the authors give the first example of a Wiener-Wintner dynamical system with continuous singular spectrum in \({\mathcal K}^\perp\). Namely, they show that any skew product on the torus of the form \(T(x,y)= (x+\alpha, y+\beta(x-[x])) \bmod 1\) is Wiener-Wintner with continuous singular spectrum in \({\mathcal K}^\perp\). In the last section of the paper, the authors prove that if \(f\in L^p\), with \(p\) large enough, then for all \(\gamma\in (1+{1\over 2p}-{\beta \over 2},1]\) there exists a set \(X_f\) of full measure for which the series \(\sum^\infty_{n=1} {f(T^nx)e^{2\pi in\varepsilon} \over n^\gamma}\) converges uniformly with respect to \(\varepsilon\).