  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.
