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The topological entropy $h_{\rm top}$ of a continuous piecewise monotoneinterval map measures the exponential growth in the number of monotonicityintervals for iterates of the map. Milnor and Thurston showed that$\exp(-h_{\rm top})$ is the smallest zero of an analytic function, now coinedthe Milnor-Thurston determinant, that keeps track of relative positions offorward orbits of critical points. On the other hand $\exp(h_{\rm top})$ equalsthe 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 leadingzero 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, thedual of the Ruelle transfer operator has a regularized determinant, identicalto the Milnor-Thurston determinant, hereby providing a natural explanation forthe above puzzle.

The Milnor-Thurston determinant and the Ruelle transfer operator