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On Bohl’s argument theorem. (English. Russian original) Zbl 1317.37033
Math. Notes 93, No. 1, 83-89 (2013); translation from Mat. Zametki 93, No. 1, 72-80 (2013).
Let \(\omega=(\omega_1,\cdots,\omega_n)\) be rationally independent, and consider the quasi-periodic motion on \(\mathbb{T}^n\) induced by \(\dot x=\omega\). Given a continuous function \(f:\mathbb{T}^n\to \mathbb{C}\), the function \(t\mapsto f(x(t))\) is called a quasi-periodic function, where \(x(t)\) is the solution with \(x(0)=x_0\) for some \(x_0\in\mathbb{T}^n\).
Bohl’s Argument Theorem. Let \(f(t)=\sigma(t) e^{i\psi(t)}\) be a quasi-periodic function, where \(\sigma(t)\geq c>0\). Then \(\psi(t)=\lambda t +g(t)\), where \(\lambda= m\cdot \omega\) for some \(m\in\mathbb{Z}^n\), and \(g(t)=G(x(t))\) for some continuous function \(G:\mathbb{T}^n\to \mathbb{R}\) with \(G(x_0)=\psi_0\). In particular, the remainder \(g\) is bounded.
In this paper, the author generalizes Bohl’s theorem to functions forced by some general flows. Let \(V(x)=(v_1,\cdots,v_n)\) be a smooth vector field on \(\mathbb{T}^n\) that preserves a smooth probability measure \(\mu\). Up to a coordinate transformation, the author shows that one can assume that \(\mu\) equals the (normalized) Lebesgue measure \(m\) on \(\mathbb{T}^n\). Let \(\omega_j=\int v_j(x) dm(x)\) be the average frequency component for each \(1\leq j\leq n\).
Let \(f\) be a smooth function on \(\mathbb{T}^n\) of unit modulus, \(x(t)\) be the solution of \(\dot x=V(x)\) with initial condition \(x(0)=x_0\), and \(\psi(t)\) be the argument function of \(f(x(t))\), that is, \(e^{i\psi(t)}=f(x(t))\). Then it is proved (Theorem 4) that if the flow \(\dot x= V(x)\) is ergodic, then for a.e. \(x_0\in\mathbb{T}^n\), \(\psi(t)=\psi_0+\lambda t+g(t,x_0)\), where \(\lambda= m\cdot \omega\) for some \(m\in\mathbb{Z}^n\), and \(g(t,x_0)=o(t)\) as \(t\to\infty\). Note that in this case, the remainder \(g(t,x)\) may not be bounded. Some sufficient conditions and counterexamples of the boundedness of \(g\) are discussed in Section 4.
MSC:
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37C10 Dynamics induced by flows and semiflows
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