Estimates of Topological Entropy of Continuous Maps with Applications

Topological entropy can be an indicator of complicated (chaotic) behavior in dynamical systems. Whether the topological entropy of a dynamical system is positive or not is of primary significance, due to the fact that positive topological entropy implies that one can assert that the system is chaotic. As the concept of topological entropy is concerned, it is hard, as remarked by [8], to get a good idea of what entropy means directly from various definitions of entropy. Thus it is enough in this paper to know that topological entropy of a dynamical system is a measure of complexity of dynamic behavior of the system, and it can be seen as a quantitative measurement of how chaotic a dynamical system is. Generally speaking, the larger the entropy of a system is, the more complicated the dynamics of this system would be. For instance, a system on a compact metric space has zero entropy provided its nonwandering set consists of finite number of periodic orbits. For the notions and discussions on entropy of dynamical systems, the reader can refer to [8, Chapter VIII]. In recent years a remarkable progress has been made in topological entropy and chaos in low-dimensional dynamical systems [1, 2, 4, 6, 7], mainly including several methods of estimating the topological entropy in one-dimensional situations. Therefore it is meaningful to present some practical results on estimating topological entropy of arbitrary dimensional dynamical systems that can be applied to real-world problems. A well-known result in chaos theory is the following theorem.