Estimates for the set of diverging points (Q1184398)

Let \(F(x,y)=(f(x)+g(x),h(x))\) be a differentiable map from \(\mathbb{R}^ 2\) to \(\mathbb{R}^ 2\) and \((F^ n)^ \infty_{n=1}\) be the corresponding dynamical system. Here \(F^ n=F\circ F\circ \dots\circ F\) (\(n\) times). Suppose that the system is dissipative, i.e., \(\text{mes}_ 2F^ n(K)\to 0\) as \(n\to \infty\) for any compact \(K\subset\mathbb{R}^ 2\). Under various additional conditions on \(F\) the authors give a description of several regions of diverging points. Remind the point \(x\) diverges under \(F\) if for every compact subset \(K\) there is a natural number \(n^ +=n^ +(K)\) such that \(F^ n(x)\) is outside \(K\) for all \(n>n^ +\).