ENTROPY OF SNAKES AND THE RESTRICTED VARIATIONAL PRINCIPLE

For interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same 'pattern' repeated over and over (one example of such orbits is what we call 'snakes'). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.