Transitive and series transitive maps on \(\mathbb{R}^d\) (Q511049)

Series transitivity is discussed, over the finite dimensional vector space \(\mathbb R^d\). The following are the main results in this paper. Theorem 1. There exists a semi-open continuous map \(f:\mathbb R^d\to \mathbb R^d\), such that i)\ \(f\) is mixing and every \(x\in \mathbb R^d\) has a dense backward orbit under \(f\) ii)\ \(f\) is series transitive. Theorem 2. Let \((X,f)\) be a homo dynamical system (i.e. \(X\) is a Polish abelian group without isolated points and \(f\) is a continuous group homomorphism) and assume that \((f-I):X\to X\) has dense range. Then, the following are equivalent: i)\ \(f\) is series transitive ii)\ there exists a dense \(G_\delta\) subset \(Z\subset X\) such that, for every \(z\in Z\), the map \(f_z:X\to X\), \(f_z(x)=f(x)+z\) is topologically transitive iii)\ there exists \(z\in X\) such that the map \(f_z\) given as before is topologically transitive iv)\ \(f\) is topologically transitive. Further aspects of these results are also discussed.