We give a uniform lower bound for the polynomial complexity of all Reeb flows on the spherization (S * M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our uniform bound is in terms of the polynomial growth of the homology of the based loops space of M . As an application, we extend the Bott-Samelson theorem from geodesic flows to Reeb flows: If (S * M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fibre S * q M , then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M is the one of a compact rank one symmetric space.Proof of (i). We will see in the proof of (ii) that if π 1 (M) has subexponential growth, then M belongs to the list in (ii), and π 1 (M) has polynomial growth of order 0, 1, 3, or 4.Proof of (ii). A main ingredient of the proof is the following Lemma 3.3. Consider a closed orientable 3-manifold M. If π 1 (M) has subexponential growth, then M admits a geometric structure modeled on one of the four geometries Ë 3 , Ë 2 × Ê, 3 , Nil .Proof. The proof can be extracted from [5], and is repeated here for the readers convenience. We distinguish several cases. Case 1: M is not prime. This means that M can be written as a connected sum M = M 1 #M 2 with both π 1 (M 1 ) and π 1 (M 2 ) non-trivial. By the Seifert-Van Kampen Theorem, π 1 (M) is the free product π 1 (M 1 ) * π 1 (M 2 ). It follows from the existence of normal forms for free products that π 1 (M 1 ) * π 1 (M 2 ) contains a free subgroup of rank 2 unless π 1 (M 1 ) = π 1 (M 2 ) = 2 , see Exercise-with-hints 19 in Sec. 4.1 of [56]. Our hypothesis on π 1 (M) thus implies π 1 (M 1 ) = π 1 (M 2 ) = 2 , and so M = ÊP 3 # ÊP 3 . This manifold has a geometric structure modeled on the geometry Ë 2 × Ê, see [79, p. 457].Case 2: M is prime, but not irreducible. Then M = S 2 × S 1 , see [42, Proposition 1.4] or [45, Lemma 3.13]. In particular, M has a geometric structure modeled on Ë 2 × Ê.Case 3: M is irreducible. We distinguish two subcases:

In this paper, we study the entropy of a Hamiltonian flow in restriction to an enregy level where it admits a first integral which is nondegenerate in the Bott sense. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies h pol and h pol * . We prove that, under conditions on the critical level of the Bott first integral and dynamical conditions on the hamiltonian function H, h * pol ∈ {0, 1} and h pol ∈ {0, 1, 2}.Institut de Mathématiques de Jussieu, UMR 7586, Analyse algébrique, 175 rue du Chevaleret, 75013 Paris. email: labrousse@math.jussieu.fr.

As we have proved in [L], the geodesic flows associated with the flat metrics on T 2 minimize the polynomial entropy h pol . In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for h pol . To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems.Institut de Mathématiques de Jussieu, UMR 7586, Analyse algébrique, 175 rue du Chevaleret, 75013 Paris. email: labrousse@math.jussieu.fr.

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.Institut de Mathématiques de Jussieu, UMR 7586, Analyse algébrique, 4, place Jussieu, 75005 Paris. email: labrousse@math.jussieu.fr.