Topological entropy
نویسندگان
چکیده
1. Definitions and general properties. Let X be a compact topological space. Definition 1. For any open cover 31 of X let N(ñ) denote the number of sets in a subco ver of minimal cardinality. A subco ver of a cover is minimal if no other subcover contains fewer members. Since X is compact and 31 is an open cover, there always exists a finite subcover. To conform with prior work in ergodic theory we call 77(31) = logAf(3l) the entropy of 31. Definition 2. For any two covers 31,33,31 v 33 = {A fïP|A£3l,P£93 } defines their jo i re. Definition 3. A cover 93 is said to be a refinement of a cover 3l,3l< 93, if every member of 93 is a subset of some member of 31. We have the following basic properties. Property 00. The operation v is commutative and associative. Property 0. The relation -< is a reflexive partial ordering (') on the family of open covers of X. Property 1.31 < 31 ',93 < 93' => 31 v 93 < 31' v93'. Proof. Consider A' n B' £ 31' v93' where A'£ 31' and P'£93'. By hypothesis there exists A £ 31 and P £ 93 such that A' ç A, B' Ç P. Thus A' n B' Q A n P where A n P £ 31v93. Remark. With the proper substitutions of 31,93 and the cover ¡Xj in the statement above we obtain 31<;3lv93 and 93 -< 3lv93 which reveals that the family of open covers is a directed set with respect to the relation -< . Property 2. 31 ■< 93 => AT(3I) ^ Ai(93), 77(31) ̂ 77(93). Proof. Let \BX, ■ ■ -,BN(ss) \ be a minimal subcover of 93. Since 3l<93. there exists a subcover \AX, ■■■,ANI^)\ of 31. Therefore Af(Sl) s' Af(93) and also 77(31) ̂ 77(93). Property 3. 3l<93 =>.rV(3lv93) = iV(93),77(31 v93) =77(93). Proof. It follows from Property 1 that 93 < 31 v93 so that AT(93) ̂ iV(3l v 93 ). On the other hand 93 >■ 31 v 93 which is a consequence of the hypothesis. Thus AT(3lv93) ^ AT(93).
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ورودعنوان ژورنال:
- Scholarpedia
دوره 3 شماره
صفحات -
تاریخ انتشار 2008