The Topological Entropy Conjecture

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