is surprisingly difficult, although a spectrum can be obtained. Borel space A vast amount of simplification results if one instead works with the Cantor space , and functions Some caution is advised, as the map is defined on the unit interval of the real number line, assuming the natural topology on the reals. By contrast, the map is defined on the Cantor space , which by convention is given a very different topology, the product topology. There is a potential clash of topologies; some care must be taken. However, as presented above, there is a homomorphism from the Cantor set into the reals; fortunately, it maps open sets into open sets, and thus preserves notions of continuity. To work with the Cantor set , one must provide a topology for it; by convention, this is the product topology. By adjoining set-complements, it can be extended to a Borel space, that is, a sigma algebra. The topology is that of cylinder sets. A cylinder set has the generic form where the are arbitrary bit values (not necessarily all the same), and the are a finite number of specific bit-values scattered in the infinite bit-string. These are the open sets of the topology. The canonical measure on this space is the Bernoulli measure for the fair coin-toss. If there is just one bit specified in the string of arbitrary positions, the measure is 1/2. If there are two bits specified, the measure is 1/4, and so on. One can get fancier: given a real number one can define a measure if there are heads and tails in the sequence. The measure with is preferred, since it is preserved by the map So, for example, maps to the interval and maps to the interval and both of these intervals have a measure of 1/2. Similarly, maps to the interval which still has the measure 1/2. That is, the embedding above preserves the measure. An alternative is to write which preserves the measure That is, it maps such that the measure on the unit interval is again the Lebesgue measure. Frobenius–Perron operator Denote the collection of all open sets on the Cantor set by and consider the set of all arbitrary functions The shift induces a pushforward defined by This is again some function In this way, the map induces another map on the space of all functions That is, given some , one