The Milnor-Thurston determinant and the Ruelle transfer operator

Abstract: The topological entropy $h_{\rm top}$ 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 $\exp(-h_{\rm top})$ 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 $\exp(h_{\rm top})$ 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.