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17 pages, 321 KiB

Open AccessArticle

by Lvlin Luo

Mathematics 2021, 9(4), 296; https://doi.org/10.3390/math9040296 - 3 Feb 2021

Cited by 1 | Viewed by 2231

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 ∂. [...] Read more.

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

{\overset{ˇ}{H}}_{p} (X; Z)

, where

0 \leq p \leq n = n_{J}

. For a continuous self-map f on X, let

α \in J

be an open cover of X and

L_{f} (α) = {L_{f} (U) | U \in α}

. Then, there exists an open fiber cover

{\dot{L}}_{f} (α)

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, {\dot{L}}_{f} (α))

through all finite open covers of

X^{f} = {L_{f} (U); U \subset 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 \in N

. Then, we prove the conjecture

log ρ \leq 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

{\overset{ˇ}{H}}_{*} (X; Z) = ⨁_{i = 0}^{n} {\overset{ˇ}{H}}_{i} (X; Z) .

Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications Methods)

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