A splitting theorem for transitive maps
The basic result is the following theorem. For a continuous transitive map \(f : X \rightarrow X\), \(X\) a locally connected compact metric space, exactly one of the following two statements holds: (1) \(f\) is totally transitive i.e. \(f^k\) is transitive for all \(k \geq 1\). (2) There exist closed connected subsets \(X_0, \dots X_{k-1}\) of \(X\), \(k > 1\), with nonempty interiors such that \(X = \bigcup_{i=0}^{k-1}X_i\), \(\text{int}(X_i) \cap X_j = \emptyset \), for \(i \neq j\), and \(f(X_i) = X_{i+1\pmod k}\). Moreover, \(f^k|_{X_i}\) is transitive, for \(i = 0, \dots, k-1\). As a consequence, the authors establish lower bounds for the topological entropy of transitive graph maps in terms of the cardinality of their splittings. They also show that for a connected graph which is not a tree, the infimum of the topological entropy of the transitive maps having a periodic point is zero.
scientific article; zbMATH DE number 1286154