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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.
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37C10 Dynamics induced by flows and semiflows
Full Text: DOI
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