Renewal Systems, Sharp-Eyed Snakes, and Shifts of Finite Type

1. INTRODUCTION. A common theme in all of mathematics is deciding when two seemingly different objects are actually the same in some sense. For example, every elementary school student knows that, despite outward appearances, the fractions 2/3 and 10/15 are the same. The calculus student knows that the functions f (x) = 3x and g(x) = (3x