Generalized recurrence and the nonwandering set for products (Q511056)

The author studies the problem of product recurrence for continuous maps on compact metrizable spaces. More precisely, if \(f:X\to X\) and \(g:Y\to Y\) are continuous maps, what can we say about recurrence properties of \(f\times g:X\times Y \to X\times Y\) given the corresponding recurrence properties of \(f\) and \(g\)? Consider for example the non-wandering set \(\mathrm{NW}(f)\) of \(f\), that is the set of points \(x\) such that, for every neighborhood \(U_x\) of \(x\) there exists \(n\in\mathbb N\) such that \(f^n(U_x)\cap U_x\neq \emptyset.\) The author shows that \(\mathrm{NW}(f\times g)\subsetneq \mathrm{NW}(f)\times \mathrm{NW}(g)\). The author also characterizes when a point \(x\in X\) is such that \((x,y)\) is nonwandering for \(f\times g\) for any \(g\) and any nonwandering point \(y\in Y\). This characterization is given in terms of \(f\) being locally mixing at \(x\). Another notion of recurrence is that of chain recurrence: a point \(x\in X\) is chain recurrent for \(f:X\to X\) if for every \(\epsilon>0\) there exist finite sequences \(x=x_0,x_1,\ldots,x_n=x\) with \(d_X(f(x_{i-1}),x)<\epsilon\), where \(d_X\) is the metric of \(X\). Denoting by \(\mathrm{CR}(f)\) the set of chain recurrent points for \(f\) the author shows that \(\mathrm{CR}(f\times g)=\mathrm{CR}(f)\times \mathrm{CR}(g)\) (chain recurrence is independent of the choice of the metrics). Similarly, a point \(x\in X\) is called strong chain recurrent for \(f:X\to X\) if for every \(\epsilon>0\) there exists a chain \(x=x_0,x_1,\ldots,x_n=x\) with \(\sum_{i=1} ^n d_X(f(x_{i-1}),x)<\epsilon\). The set of strong recurrent points is denoted by \(\mathrm{SCR}_{d_X}(f)\) and it depends on the metrics \(d_X\). One gets rid of this dependence by considering \(\mathrm{GR}(f):=\cap_{d' _X}\mathrm{SCR}_{d' _X}(f)\), the intersection ranging over all metrics \(d' _X\) compatible with the topology in \(X\). Then the author shows that \(\mathrm{GR}(f\times g)=\mathrm{GR}(f)\times \mathrm{GR}(g)\). The relation between the recurrent sets of \(f,g\) and the recurrent set of \(f\times g\) is studied for several other notions of recurrence. The results above assume that one considers metrics \(d_{X\times Y}\) compatible with the product topology.