A homological bound on entropy in arbitrary compact spaces
Luis Hernández–Corbato, David Jesús Nieves-Rivera, Francisco Romero Ruiz del Portal, J. J. Sánchez-Gabites · Mathematische Zeitschrift · 2025
Abstract A result of Manning states that for a compact manifold X and a continuous map $$f: X \rightarrow X$$ f : X → X the topological entropy of f is bounded below by the logarithm of the spectral radius of the map induced by f in the first homology group $$H_1(X;\mathbb {C})$$ H 1 ( X ; C ) . We generalize this result to arbitrary compact spaces X in terms of Čech cohomology $$\check{H}^1(X;\mathbb {C})$$ H ˇ 1 ( X ; C ) . The essential tool is a notion of integration of Alexander-Spanier cocycles over Čech cycles. Most of the discussion is carried out “at scale $$\mathcal {U}$$ U ”, for an open covering $$\mathcal {U}$$ U . This is used to keep track of the “homological length” of the iterates of a cycle, which in turn leads to a lower bound on the number of elements in the coverings that appear in the definition of the entropy.