Abstract
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 nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(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 Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).
Highlights
IntroductionIn 1980, Katok [19] proved that if a C1+α (α > 0) diffeomorphism f of a compact manifold has a Borel probability continuous (non-atomic) invariant ergodic measure with non-zero Lyapunov exponents, it has positive topological entropy
Recall that the pair (X, f ) is called a topological dynamical system, which is induced by the iteration: fn = f ◦···◦ f, n ∈ N n and f 0 is denoted the identity self-map on X, where X is a compact Hausdorff space and f is a continuous self-map on X
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 nJ, where nJ is the dimension of X associated with ∂, which will become clear in Definition 3
Summary
In 1980, Katok [19] proved that if a C1+α (α > 0) diffeomorphism f of a compact manifold has a Borel probability continuous (non-atomic) invariant ergodic measure with non-zero Lyapunov exponents, it has positive topological entropy. In 1992, for n-dimensional compact Riemannian manifolds with n ∈ Z+, Paternain made a relation between the geodesic entropy and topological entropy of the geodesic flow on the unit tangent bundle [23], which is an improvement of Manning’s inequality [24]. In 2006, Zhu [30] showed that for Ck-smooth random systems, the volume growth is bounded from above by the topological entropy on compact Riemannian manifolds. In 2019, Lima et al [36] developed symbolic dynamics for smooth flows with positive topological entropy on three-dimensional closed (compact and boundaryless) Riemannian manifolds. For a variational principle for subadditive preimage topological pressure for continuous bundle random dynamical systems, see [42]
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