Computing the topological entropy for piecewise monotonic maps on the interval (Q1308026)

The topological entropy of a map is one of the quantitative measures of the complexity of a dynamical system. This paper deals with a new method for computing the topological entropy of a piecewise monotonic transformation on the interval. For a piecewise monotonic map \(T\) on the interval one can define the topological entropy \(h_{\text{top}}(T)\) by \[ h_{\text{top}}(T): =\lim_{n\to \infty}{1\over n}\log \bigl(c_n(T)\bigr). \] The author shows that \(c_n(T)=c_ne^{h_{\text{top}}(T)\cdot n} +r_n\), where the \(c_n\) are bounded and periodic and \(|r_n|\leq K\alpha^n\) for a constant \(K\) and \(\alpha<e^{h_{\text{top}}(T)}\). The proof of this fact is based on a transition matrix associated with \(T\). For this matrix the author gives a spectral theorem. This is used for an estimation of the accuracy of an algorithm.