| Summary: | For a compact Hausdorff space <i>X</i>, let <i>J</i> be the ordered set associated with the set of all finite open covers of <i>X</i> such that there exists <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>n</mi><mi>J</mi></msub></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>n</mi><mi>J</mi></msub></semantics></math></inline-formula> is the dimension of <i>X</i> associated with <i>∂</i>. Therefore, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo>ˇ</mo></mover><mi>p</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>n</mi><mo>=</mo><msub><mi>n</mi><mi>J</mi></msub></mrow></semantics></math></inline-formula>. For a continuous self-map <i>f</i> on <i>X</i>, let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi>J</mi></mrow></semantics></math></inline-formula> be an open cover of <i>X</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mi>L</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>|</mo><mi>U</mi><mo>∈</mo><mi>α</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Then, there exists an open fiber cover <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>L</mi><mo>˙</mo></mover><mi>f</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mi>f</mi></msup></semantics></math></inline-formula> induced by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we define a topological fiber entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> as the supremum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mi>n</mi><mi>t</mi><mo>(</mo><mi>f</mi><mo>,</mo><msub><mover accent="true"><mi>L</mi><mo>˙</mo></mover><mi>f</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> through all finite open covers of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>X</mi><mi>f</mi></msup><mo>=</mo><mrow><mo>{</mo><msub><mi>L</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>;</mo><mi>U</mi><mo>⊂</mo><mi>X</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the f-fiber of <i>U</i>, that is the set of images <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mi>n</mi></msup><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and preimages <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>. Then, we prove the conjecture <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mi>ρ</mi><mo>≤</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <i>f</i> being a continuous self-map on a given compact Hausdorff space <i>X</i>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is the maximum absolute eigenvalue of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mo>*</mo></msub></semantics></math></inline-formula>, which is the linear transformation associated with <i>f</i> on the Čech homology group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo>ˇ</mo></mover><mo>*</mo></msub><mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munderover><mo>⨁</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover></mstyle><mrow><msub><mover accent="true"><mi>H</mi><mo>ˇ</mo></mover><mi>i</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow></mrow><mo>.</mo></mrow></semantics></math></inline-formula>
|
|---|
