CALCULATING TOPOLOGICAL-ENTROPY

The attempt to find effective algorithms for calculating the topologic al entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem. The algor ithm introduced here is motivated by the fact that if f: [0, 1] --> [0 , 1] is a piecewise monotone map of the unit interval into itself, the n h(f) = lim(n-->infinity) (1/n) log Var(f(n)), where h(f) is the topo logical entropy off; and Var(f(n)) is the total variation of f(n). We show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function, because of the slow convergence. However, a close examination of the reasons for this failure leads ultimately to the modified algorithm wh ich is presented in this paper. We prove that this algorithm is equiva lent to the standard power method for finding eigenvalues of matrices (with shift of origin) in those cases for which the Function is Markov , and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one-parameter Famil ies of test functions.