Dynamical Zeta Functions for Tree Maps

: We study piecewise monotone and piecewise continuous maps f from a rooted oriented tree to itself, with weight functions either piecewise constant or of bounded variation. We define kneading coordinates for such tree maps. We show that the Milnor-Thurston relation holds between the weighted reduced zeta function and the weighted kneading determinant of f . This generalizes a result known for piecewise monotone interval maps. 1 Introduction Let f : [0; 1] ! [0; 1] be strictly monotone on the subintervals defined by the turning points 0 = a 0 ! a 1 ! : : : ! aN = 1 (with N finite). Let g : [0; 1] ! j C be a weight. We denote by f ffin the map f ffi f ffi : : : ffi f composed n times. If each iterate of f has finitely many fixed points, it is natural to define the weighted Ruelle zeta function : i f;g (t) = exp X n1 t n n X x:f ffin x=x n\Gamma1 Y k=0 g(f ffik x) : If g j 1 we simply have the unweighted zeta function i f (t) = exp X n1 t n n #fx : f ffin x =..