Abstract
The aim of this paper is to state and prove a polynomial analogue of the classical Manning inequality, relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim, we use a numerical conjugacy invariant for dynamical systems, the polynomial entropy. It is infinite when the topological entropy is positive. We first prove that the growth rate of the volume of balls is bounded above by means of the polynomial entropy of the geodesic flow. For the flat torus this inequality becomes an equality. We then study explicitely the example of the torus of revolution (which is a case of strict inequality). We give an exact asymptotic equivalent of the growth rate of volume of balls.
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