A homological bound on entropy in arbitrary compact spaces

Abstract A result of Manning states that for a compact manifold X and a continuous map $$f: X \rightarrow X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> 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 $$H_1(X;\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We generalize this result to arbitrary compact spaces X in terms of Čech cohomology $$\check{H}^1(X;\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>ˇ</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . The essential tool is a notion of integration of Alexander-Spanier cocycles over Čech cycles. Most of the discussion is carried out “at scale $$\mathcal {U}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> ”, for an open covering $$\mathcal {U}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> . 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.