Unique ergodicity of dynamic systems on the torus
The authors consider a dynamical system on the torus \(T^2\) defined by \[ \frac{dx}{dt}=G(x,y),\quad \frac{dy}{dt}=H(x,y), \] where \(G\) and \(H\) are continuous functions of period 1 in \(x\) and \(y\). In the region \(0\leq x<1\), \(0\leq y<1\), the only singular point is \(x=0\), \(y=0\). Let \(f\) denote the \(C^1\)-flow associated with this system. The authors prove that when the rotation number of \(f\) is either rational or infinite, then \(f\) is uniquely ergodic if and only if \(f\) has no periodic orbit. They also show that an \(f\)-invariant probability measure \(m\) on \(T^2\) is a nontrivial ergodic measure if and only if the rotation number of \(f\) is irrational and \(m(A)=0\) where \(A\) consists of the singular point and all other points whose \(\alpha\)- or \(\omega\)-limit set is the singular point. Finally, the authors demonstrate that \(f\) has at most one non-trivial ergodic measure.
scientific article; zbMATH DE number 987495