The Topological Entropy Conjecture

For a compact Hausdorff space X , let J be the ordered set associated with the set of all finite open covers of X such that there exists n J , where n J is the dimension of X associated with ∂ . Therefore, we have H ˇ p ( X ; Z ) , where 0 ≤ p ≤ n = n J . For a continuous self-map f on X , let α ∈ J be an open cover of X and L f ( α ) = { L f ( U ) | U ∈ α } . Then, there exists an open fiber cover L ˙ f ( α ) of X f induced by L f ( α ) . In this paper, we define a topological fiber entropy e n t L ( f ) as the supremum of e n t ( f , L ˙ f ( α ) ) through all finite open covers of X f = { L f ( U ) ; U ⊂ X } , where L f ( U ) is the f-fiber of U , that is the set of images f n ( U ) and preimages f − n ( U ) for n ∈ N . Then, we prove the conjecture log ρ ≤ e n t L ( f ) for f being a continuous self-map on a given compact Hausdorff space X , where ρ is the maximum absolute eigenvalue of f * , which is the linear transformation associated with f on the Čech homology group H ˇ * ( X ; Z ) = ⨁ i = 0 n H ˇ i ( X ; Z ) .