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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 $\langle F\rangle=\int F(x)f(x)\,dx_1dx_2,\quad \Lambda={1\over4\pi^2}\int f(x)\,dx_1dx_2.$

##### MSC:
 37A25 Ergodicity, mixing, rates of mixing 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 60J60 Diffusion processes
##### Keywords:
diffusion process; torus; integral invariant