Polynomial growth of the volume of balls for zero-entropy geodesic systems

Labrousse, Clémence

doi:10.1088/0951-7715/25/11/3049

Cited by 20 publications

(8 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are various ways of defining h top (ϕ), see [40]. If one replaces in these definitions the denominator m by log m, one obtains the slow entropy slow-h top (ϕ), an invariant introduced in [59] (see also [50]) and further studied in [52,53,54]. The invariants slow-vol(ϕ) and slow-h top (ϕ) do not always agree, however.…”

Section: Let ϕ Tmentioning

confidence: 99%

Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

2015

Self Cite

View full text Add to dashboard Cite

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:

show abstract

Section: Let ϕ Tmentioning

confidence: 99%

Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

2015

Self Cite

View full text Add to dashboard Cite

show abstract

“…In order to characterize the flat metrics, it is therefore useful to consider a finer dynamical invariant of the geodesic flow, such as the polynomial entropy, introduced in [22]. Using the techniques of [22], it was proved in [17] that the polynomial entropy of a flat torus of dimension d (in restriction to the sphere bundle) is equal to d − 1, which is a lower bound for the polynomial entropy of all metrics on T d . It was also proved in [19] that the polynomial entropy of the revolution two torus is two, which is higher than the one of the flat two tori.…”

Section: Introductionmentioning

confidence: 99%

“…Using the techniques of [20], it was proved in [16] that the polynomial entropy of a flat torus of dimension d (in restriction to the sphere bundle) is equal to d − 1, which is a lower bound for the polynomial entropy of all metrics on T d . It was also proved in [18] that the polynomial entropy of the revolution two torus is two, which is higher than the one of the flat two tori.…”

Section: Introductionmentioning

confidence: 99%

An entropic characterization of the flat metrics on the two torus

Bernard

Labrousse

2015

Geom Dedicata

Self Cite

View full text Add to dashboard Cite

The geodesic flow of the flat metric on a torus is minimizing the polynomial entropy among all geodesic flows on this torus. We prove here that this properties characterises the flat metric on the two torus.Résumé Le flot géodésique des métriques plates sur un tore minimise l'entropie polynomiale parmi tous les flots géodésique sur ce tore. On montre ici que cette propriété caractérise les métriques plates en dimension deux.

show abstract

“…This quantity must be taken in R + ∪ {+∞}, but it will be finite for most of the systems we shall consider. The polynomial entropy has already been studied in several contexts: for integrable Hamiltonian systems by Marco [38,39], for Brouwer homeomorphisms by Hauseux and Le Roux [19], and in various geometric situations by Bernard, Labrousse and Marco [3,27,28,29,30]. A similar notion was defined by Katok and Thouvenot, see [25] and [22,24].…”

mentioning

confidence: 99%

Automorphisms of compact Kähler manifolds with slow dynamics

Cantat¹,

Paris-Romaskevich²

2020

Preprint

View full text Add to dashboard Cite

We study the automorphisms of compact Kähler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions 2 and 3. We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the C ∞ context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension 2 and prove they exist only on tori. We conjecture that this is true for any dimension.1.2. Polynomial entropy. Let (X, d) be a compact metric space and f : X → X a continuous map. The Bowen metric, at time n for the map f , is the distance

show abstract

Polynomial growth of the volume of balls for zero-entropy geodesic systems

Cited by 20 publications

References 16 publications

Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

An entropic characterization of the flat metrics on the two torus

Automorphisms of compact Kähler manifolds with slow dynamics

Contact Info

Product

Resources

About