2.5. Topological entropy

Another dynamical entropy can be evaluated for a mathematical system. This is the entropy [20,21,22], H{T\ which gives a measure of the number of distinct trajectories generated b> a system. Unlike the metric entropy, h(/i, T), the topological entropy is a property of T alone and is not associated with any metric properties, of the dynamics. It provides a measure of the number of trajectories, or orbits, {x, Tx, T2x ...}T has. This appears to be infinite, like the number of choices for jc. However, crb'ts {jc, Tx, T2x...} and {y, 7y, T2y...} are only considered distinct if [21]12

 

 

If one Observes up to the nth iterate there will now only exist m enumerable number ©I orbits. Now if M(g, n) is the maximum number of different orbits (that is trajectories separated by greater than «) of length n, the topological entropy is defined as a measure of the exponential growth of M with n in the limit of arbitrarily fine discrimination between trajectories

 

 

So, roughly, this indicates M ~ exp(Hn) in the limit. T».n topological entropy gives the rate of growth of orbits with finite length as their allowed length goes to infinity («->») and resolution fidelity becomes arbitrarily fine (e -* 0). As an example consider a system which prints out a sequence of symbols using a basic alphabet of m different characters. The number of admissible character strings of length n is mn, e is the length of a sequence examined and H = log m.

If a system contained a si. gle attracting fixed point (for example, a strongly damped harmonic oscillator), then all trajectories would wind into this point as the stationary solution was approached. Eventually, all trajectories will be closer to each other than any pre-ordaine d e. Therefore, in the sense of (2.38) this system has only one distinct orbit and H = 0. Remarkably, some systems have H >0 and these are called chaotic.

It can also be shown that the topological entropy is determined [21] by the number of fixed pointst of T", and the following is equivalent to (2.39),

 

 

There is close relationship between Kolmogorov's metric entropy, h{,i, T), and the topological entropy. In particular, if T preserves several finite invariant measures thei a metric entropy ht{jih T) can be associated with each. The topologic.' ntropy can be shown to be equal to the largest metric entropy and the corresponding maximal measure is referred to as the Gibbs measure,

 

 

If T preserves a unique invariant measure then h and H will be equal.

One of the important features of both the metric and toooloeical entroDies is their stable character. If one slightly titers some parameter determining the evolution of a chaotic dynamical system, "arge change in its behaviour will generally result because oi its exponenthl traj ctory instability. However, the above-me ntioned dynamical entropies are often slowly varying with respect to such perturbations. YLey are the preferred means of stable description.