Let \(G\) be a locally compact group and consider the Orlicz space \(L^\Phi(G)\), defined by means of a Young function \(\Phi\) through the usual Luxemburg norm. The author studies recurrence and chaoticity properties of weighted shifts \(T_{a,w}f(x):= w(x) f(xa^{-1})\), where \(w:G\to(0,\infty)\) is a weight.
An operator \(T\) acting on a (separable Banach) space \(X\) is called topologically multiply recurrent if, for every positive integer \(L\) and every nonempty \(U\subset X\), there exists \(n\in\mathbb N\) such that \(U\cap T^{-n}U\cap\cdots\cap T^{-Ln}U\neq \emptyset\). On the other hand, \(T\) is said to be chaotic if it has a dense set of periodic points and there exists \(x\in X\) whose orbit under \(T\), that is, \(\mathrm{Orb}(x,T)=\{T^nx: n\in\mathbb N\}\), is dense in \(X\) (namely, \(T\) is hypercyclic and has a dense set of periodic points).
The author gives a characterization of topologically multiply recurrent weighted shifts \(T_{a,w}\) on \(L^\Phi(G)\) by means of a rather technical condition involving \(a\) and \(w\). The author also provides a characterization of chaoticity by means of a condition on \(a,w\). It turns out that \(T_{a,w}\) is chaotic exactly when its set of periodic elements is dense (certain elements \(a\in G\) have to be a priori excluded; these are the periodic elements of \(G\) which generate a compact subgroup \(G(a)\)). A consequence of these results is that, if \(a\in G\) is a periodic and the weighted shift \(T_{a,w}\) is chaotic on \(L^\Phi(G)\), then it is topologically multiply recurrent.