Original Definitions

Original definitions of redundancy came from information theory, long before chaos came along. Those definitions aren't exactly the same as the one used in chaos theory. In other words, the idea of redundancy as mutual information isn't the only concept of "redundancy" in the literature. However, most schools of thought, within both information theory (e.g. Shannon & Weaver 1949: 56) and chaos theory (e.g. Gershenfeld & Weigend 1993: 48), define redundancy in ways that are closely related to the "chaotic" ones. For background, let's take a quick look at how information theory usually defines redundancy. There are two ways. Each can be derived from the other.
The first definition is:

redundancy = the difference between some reference entropy and an actual or observed entropy. ......(1)

Depending on the circumstances or problem, we can choose different definitions for "actual entropy" and "reference entropy." For instance, for one system or variable, redundancy might be the difference between the maximum possible entropy (the reference entropy) and the actual or self-entropy, say H_X. Maximum possible entropy occurs when the probability of every bin is the same; in that case it equals logN_s, where N_s is the number of possible states or bins. Redundancy by the above definition then would be the difference between the reference entropy (here logN_s) and self-entropy (here H_X), or (logN_s)-H_X.
The second definition is a standardized version of the first definition. The standardization is done by dividing the basic difference by the reference entropy. That puts all redundancies within the range of 0 to 1. In the example just used, standardized redundancy becomes definition one divided by the reference entropy, or (logN_s- H_X)/logN_s. That ratio, according to the rules of basic algebra, is (logNs/logNs)-(HX/logNs), or 1-(H_X/logN_s). Thus, the second definition of redundancy as used in information theory is:

redundancy = 1 -(actual entropy/reference entropy).

The ratio of the actual entropy to the reference entropy is a type of relative entropy (the ratio of one entropy to another).