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

2015 ◽  
Vol 07 (03) ◽  
pp. 407-451 ◽  
Author(s):  
Urs Frauenfelder ◽  
Clémence Labrousse ◽  
Felix Schlenk
Keyword(s):  
Lower Bound ◽  
Polynomial Growth ◽  
Polynomial Complexity ◽  
Cohomology Ring ◽  
Volume Growth ◽  
Universal Cover ◽  
Geodesic Flows ◽  
Dynamical Complexity ◽  
Reeb Flows ◽  
Rank One

We give a uniform lower bound for the polynomial complexity of 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 lower bound is in terms of the polynomial growth of the homology of the based loop 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 fiber [Formula: see text], then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M agrees with that of a compact rank one symmetric space.

scholarly journals C0-stability of topological entropy for contactomorphisms

2021 ◽  
pp. 2150015
Author(s):  
Lucas Dahinden
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lower Bound ◽  
Topological Entropy ◽  
Small Perturbation ◽  
Geodesic Flows ◽  
Reeb Flows ◽  
Lower Semi ◽  
Special Classes

Topological entropy is not lower semi-continuous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive contactomorphisms) topological entropy at least is stable in the sense that there exists a nontrivial continuous lower bound, given that a certain homological invariant grows exponentially.

scholarly journals Average of Uncertainty Product for Bounded Observables

2018 ◽  
Vol 25 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Lin Zhang ◽  
Jiamei Wang
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Quantum States ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Uncertainty Product ◽  
Schmidt Norm ◽  
Rank One ◽  
Pure States ◽  
Average Uncertainty

The goal of this paper is to calculate exactly the average of uncertainty product of two bounded observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Here we use the uniform ensembles of pure and isospectral states as well as the states distributed uniformly according to the measure induced by the Hilbert-Schmidt norm. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. These results can help us check how large is the gap between the uncertainty product and any lower bounds obtained for the uncertainty product. Although our method in the present paper cannot give a tighter lower bound of uncertainty product for bounded observables, it can help us drop any one that is not substantially tighter than the known one.

scholarly journals Pressures for geodesic flows of rank one manifolds

Nonlinearity ◽  
2014 ◽  
Vol 27 (7) ◽  
pp. 1575-1594 ◽  
Author(s):  
Katrin Gelfert ◽  
Barbara Schapira
Keyword(s):  
Geodesic Flows ◽  
Rank One

scholarly journals Cocycle rigidity of partially hyperbolic abelian actions with almost rank-one factors

2017 ◽  
Vol 39 (7) ◽  
pp. 2006-2016
Author(s):  
KURT VINHAGE
Keyword(s):  
Recent Progress ◽  
Universal Cover ◽  
Cycle Structure ◽  
Rank One ◽  
Partially Hyperbolic ◽  
Cocycle Rigidity

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank-one factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.

Some regularity estimates for convolution semigroups on a group of polynomial growth

2004 ◽  
Vol 77 (2) ◽  
pp. 249-268 ◽  
Author(s):  
Nick Dungey
Keyword(s):  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Kernels ◽  
High Order ◽  
Convolution Semigroup ◽  
Geometric Condition ◽  
Convolution Semigroups ◽  
Regularity Estimates ◽  
Convolution Powers ◽  
Gaussian Estimates

AbstractWe study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

scholarly journals Dimensional Bounds for Ancient Caloric Functions on Graphs

2019 ◽  
Author(s):  
Bobo Hua
Keyword(s):  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Heat Equations ◽  
Ancient Solutions

Abstract We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi’s theorem [9] on manifolds to graphs: for a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.

scholarly journals Volume growth in the component of fibered twists

2018 ◽  
Vol 20 (08) ◽  
pp. 1850014 ◽  
Author(s):  
Joontae Kim ◽  
Myeonggi Kwon ◽  
Junyoung Lee
Keyword(s):  
Lower Bound ◽  
Symplectic Manifold ◽  
Infinite Order ◽  
Floer Homology ◽  
Volume Growth ◽  
Spectral Sequences ◽  
Connected Component ◽  
Text Type ◽  
Infinite Dimensional ◽  
Homology Groups

For a Liouville domain [Formula: see text] whose boundary admits a periodic Reeb flow, we can consider the connected component [Formula: see text] of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, in the component [Formula: see text] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [Formula: see text] has infinite order in [Formula: see text] if there is an admissible Lagrangian [Formula: see text] in [Formula: see text] whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of [Formula: see text]-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse–Bott spectral sequences.

scholarly journals Positive topological entropy of Reeb flows on spherizations

2011 ◽  
Vol 151 (1) ◽  
pp. 103-128 ◽  
Author(s):  
LEONARDO MACARINI ◽  
FELIX SCHLENK
Keyword(s):  
Topological Entropy ◽  
Energy Surface ◽  
Klein Bottle ◽  
Closed Surface ◽  
Hyperbolic Manifolds ◽  
Geodesic Flows ◽  
Hamiltonian Flow ◽  
Reeb Flows ◽  
Closed Orbits ◽  
Geometric Idea

AbstractLet M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H.Theorem 1. The topological entropy of ϕt is positive.This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive.For q ∈ M abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points.Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time.In the lowest dimension, Theorem 1 yields the existence of many closed, orbits.Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

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