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Diffusion in systems with an integral invariant on a torus. (English. Russian original) Zbl 1044.37004
Dokl. Math. 64, No. 3, 390-392 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 381, No. 5, 596-598 (2001).
Differential equations on the two-dimensional torus \(T^2=\{(x,y)\pmod2\pi\}\) with integral invariant and without equilibrium positions reduce to the form \[ \dot x_1={\omega_1\over f},\quad\dot x_2={\omega_2\over f},\tag{1} \] where \(f\) is a smooth positive function.
Kolmogorov proved if \(\gamma={\omega_1\over\omega_2}\) is “poorly” approximated by rationals (that is, for almost all \(\gamma\)), (1) can be reduced to an equation with \(f\) equals constant \(\Lambda\).
Definition: Let \(g^t\) be the phase flow of (1), and \(F\) and \(G\) belong to \(L^2\) . Then system (1) is called a diffusion system if \[ K(t)=\int F(g^{-t}(x))G(x)f(x)\,dx_1dx_2 \] has a limit as \(t\to\pm\infty\).
If \(\gamma\) is rational, system (1) is nonergodic. However, diffusion may occur. When \(\gamma\) is rational, system (1) can be reduced to \[ \dot v_1={\Omega\over \lambda(v_2)},\quad \dot v_2=0. \] For a more general system \[ \dot x_1=\omega(y),\quad \dot y=0, \] the following theorem holds:
If all critical points of a function \(y\mapsto\omega(y)\) are nongenerate, then \[ \lim_{t\to\infty}K(t)=2\pi\int_0^{2\pi}ab\,dy, \] where \[ a(y)={1\over2\pi}\int_0^{2\pi}F(x,y)\,dx,\quad b(y)={1\over2 \pi}\int_0^{2\pi}G(x,y)\,dx. \] Thus, for system (2), the following theorem holds:
Suppose that \(\gamma\) is rational and all critical points of the periodic function \(\lambda(\cdot)\) are nondegenerate. Suppose also that \(F\) and \(G\) are the characteristic functions of measureable domains \(X\) and \(Y\) of positive measure on the torus, and almost all trajectries of system (1) intersect \(X\). Then \[ \lim_{t\to\infty}K(t)>0. \] Now, fix \(L^2\) functions \(F\) and \(G\). Consider a sequence of rational numbers \(\{\gamma_n\}\) tending towards an irrational \(\gamma\). Let \(K_n(t)\) and \(\lambda_n\) be the one corresponding to \(\gamma_n\). Denote \(\kappa_n=\lim_{t\to\infty}K_n(t)\).
Then the following theorem holds: If \(\lambda_n\) are Morse functions at all \(n\), then \[ \lim_{n\to\infty}\kappa_n={\langle F\rangle\langle G\rangle\over \Lambda}, \] where \[ \langle F\rangle=\int F(x)f(x)\,dx_1dx_2,\quad \Lambda={1\over4\pi^2}\int f(x)\,dx_1dx_2. \]

37A25 Ergodicity, mixing, rates of mixing
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
60J60 Diffusion processes