×

Directional entropy of rotation sets. (English. Abridged French version) Zbl 0985.37006

Let \((X,T)\) be a compact topological dynamical system. Any \(n\)-tuple \(f= (f_1,\dots, f_n)\) of continuous real-valued maps on \(X\) induces a map \(\mu\mapsto f_*(\mu)= (\int f_1 d\mu,\dots,\int f_n d\mu)\) from the set \({\mathcal M}\) of \(T\)-invariant probability measures on \(X\) to \(\mathbb{R}^n\), and the set \(f_*({\mathcal M})\) is called the rotation set of \(f\). Following Geller and Misiurewicz, a measure \(\mu\in{\mathcal M}\) is called directional if \(f_*(\nu)= f_*(\mu)\) for every \(\nu\in{\mathcal M}\) whose support is contained in that of \(\mu\), and lost otherwise.
The author investigates certain geometric properties of \(f_*({\mathcal M})\) under additional hypotheses on \((X,T)\). The main result (Theorem 2) states that, if \((X,T)\) is a mixing shift of finite type, and if the components of \(f\) are cohomologically independent functions of summable variation, then any \(\rho\) which is either an interior or an exposed extremal point of \(f_*({\mathcal M})\) has the property that \[ \sup_{\substack{ \nu\in f^{-1}_*(\rho)\\ \nu\text{ is directional}}} h_\nu(T)= \sup_{\nu\in f^{-1}_*(\rho)} h_\nu(T), \] where \(h_\nu(T)\) denotes metric entropy. If \(\rho\) is interior, then this supremum is attained by a unique lost measure in \(f^{-1}_*(\rho)\), and if \(\rho\) is exposed, the supremum is attained by at least one ergodic and directional measure, but not by any lost measure.
The author gives no information about non-exposed extremal points in \(f_*({\mathcal M})\).

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
PDFBibTeX XMLCite
Full Text: DOI