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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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##### References:
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