Open dynamics on subshifts of finite type

Abstract: Dynamical systems can be broadly classified into closed and open systems. In a (traditional) closed system, the orbit of a point lies in the state space for all time, whereas in an open system, the orbit of a point may eventually escape from the state space through a hole. The notion of open dynamical systems was introduced by Pianigiani and Yorke in 1979, motivated by the dynamics of a ball on a billiard table with pockets. It has attracted the attention of researchers since then especially due to its wide applications. In this talk, we consider an important class of dynamical systems known as the subshifts of finite type. We study the average rate at which the orbits escape into the hole (termed as the escape rate). This problem turns out to be an interesting application of a combinatorial question of counting the number of words of given length not containing any of the words from a fixed collection as subwords. Using this, we compare the escape rates into different holes. We also present an application of our results in computing the Perron eigenvalues and eigenvectors of any non-negative integer matrix.