Algebraic topology: a first course by Marvin J. Greenberg

By Marvin J. Greenberg

Great first e-book on algebraic topology. Introduces (co)homology via singular theory.

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Example text

In the case of Ý is defined quite separately to Ý , the set the size and number of covers used to determine the box dimension. Further, it is Ý often the case that sets of the form of Ý are dense, in which case supp . 2) dimensions respectively. 5 Definition 1. 9 respectively, exist and satisfy the Legendre transform pair ´µ ´µ ´Õµ ÒÝ ÕÝ ´Ýµ ´Ýµ ÒÕ ÕÝ ´Õ µ and 2. 11) ´ µ ´ µ ´ µ ´µ ´µ ´µ Ý and also between Notice the difference between the sets Ò Ý ¯ and Ý ¯ Ý ¯ and Ý . The sets and Ý ¯ describe the local behaviour in Ò Æ Æ a pre-limiting sense, while Ý and Ý describe the limiting local behaviour.

This theory arises in connection with the transmission of information, in particular, the length of a Æ, binary representation of that information. Say a set has Ò elements. If Ò · where Æ ¾ , each element can be labelled by a binary number having Æ digits. As such, Hartley defined . ¾ Ò as the necessary information to characterise Now assume that ¡¡¡ , where ½ ¡ ¡ ¡ are pairwise ½ ¾ disjoint finite sets. An experiment is performed, which consists of independently and randomly allocating the Ò elements to the subsets ( , ¡ ¡ ¡ ) according to the probabilities Ô .

1 is invariant to the way that the probability is allocated. It is these R´enyi dimensions that change according to the manner in which the probability is allocated within the Cantor set. The second context is that the rescaled cumulant generating function is a type of global average and occurs in the theory of large deviations, which will be used in Part II of the book. The global averaging or R´enyi dimensions, given by Õ or Õ , respectively, are often related to the multifractal spectrum, Ý , by a Legendre transform.

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