The Milnor–Thurston Determinant and the Ruelle Transfer Operator

Abstract

The topological entropy h _top of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iterates of the map. Milnor and Thurston showed that exp(–h _top) is the smallest zero of an analytic function, now coined the Milnor–Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand exp(h _top) equals the spectral radius of a Ruelle transfer operator L, associated with the map. Iterates of L keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M–T determinant but all peripheral (those lying in the unit disk) zeros are the same. The purpose of this note is to show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor–Thurston determinant, hereby providing a natural explanation for the above puzzle.