scholarly journals Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one

2018 ◽  
Vol 38 (1) ◽  
pp. 343-361 ◽  
Author(s):  
Nasab Yassine ◽  
Keyword(s):  
Dynamical Systems ◽  
Infinite Measure ◽  
Quantitative Recurrence ◽  
Dimension One

Mixing for invertible dynamical systems with infinite measure

2015 ◽  
Vol 15 (02) ◽  
pp. 1550012 ◽  
Author(s):  
Ian Melbourne
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Large Class ◽  
Renewal Theory ◽  
Additional Structure ◽  
Infinite Measure ◽  
Higher Order Asymptotics ◽  
Mixing Rates ◽  
Noninvertible Maps ◽  
Invent Math

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math.189 (2012) 61–110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure. Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates and higher order asymptotics.

scholarly journals PREDICTING PATTERN FORMATION IN PARTICLE INTERACTIONS

2012 ◽  
Vol 22 (supp01) ◽  
pp. 1140002 ◽  
Author(s):  
JAMES H. VON BRECHT ◽  
DAVID UMINSKY ◽  
THEODORE KOLOKOLNIKOV ◽  
ANDREA L. BERTOZZI
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Ground State ◽  
Self Assembly ◽  
Systems Approach ◽  
Particle Interactions ◽  
Space Filling ◽  
System Of Particles ◽  
Large Systems ◽  
Dimension One

Large systems of particles interacting pairwise in d dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere (in 3D) or a ring (in 2D). Localized, space-filling, co-dimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a nonlocal linear stability analysis for particles uniformly distributed on a d - 1 sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in self-assembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

scholarly journals A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials

2010 ◽  
Vol 31 (4) ◽  
pp. 1043-1071 ◽  
Author(s):  
VÍTOR ARAÚJO ◽  
ALEXANDER I. BUFETOV
Keyword(s):  
Dynamical Systems ◽  
Large Deviations ◽  
Moduli Space ◽  
Geodesic Flow ◽  
Large Deviation ◽  
Suspension Flows ◽  
Abelian Differentials ◽  
Quantitative Recurrence ◽  
Teichmuller Geodesic Flow ◽  
Teichmüller Flow

AbstractLarge deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems. Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata119 (2006), 121–140].

Improved mixing rates for infinite measure-preserving systems

2013 ◽  
Vol 35 (2) ◽  
pp. 585-614 ◽  
Author(s):  
DALIA TERHESIU
Keyword(s):  
Dynamical Systems ◽  
Simple Proof ◽  
Renewal Theory ◽  
New Technique ◽  
First Order ◽  
Return Map ◽  
Infinite Measure ◽  
Mixing Rates ◽  
A New Technique ◽  
Invent Math

AbstractIn this work, we introduce a new technique for operator renewal sequences associated with dynamical systems preserving an infinite measure that improves the results on mixing rates obtained by Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61–110]. Also, this technique allows us to offer a very simple proof of the key result of Melbourne and Terhesiu that provides first-order asymptotics of operator renewal sequences associated with dynamical systems with infinite measure. Moreover, combining techniques used in this work with techniques used by Melbourne and Terhesiu, we obtain first-order asymptotics of operator renewal sequences under some relaxed assumption on the first return map.

scholarly journals Operator renewal theory for continuous time dynamical systems with finite and infinite measure

2016 ◽  
Vol 182 (2) ◽  
pp. 377-431 ◽  
Author(s):  
Ian Melbourne ◽  
Dalia Terhesiu
Keyword(s):  
Dynamical Systems ◽  
Continuous Time ◽  
Renewal Theory ◽  
Infinite Measure

scholarly journals Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure

2018 ◽  
Vol 12 (1) ◽  
pp. 285-313
Author(s):  
Ian Melbourne ◽  
◽  
Dalia Terhesiu ◽  
Keyword(s):  
Dynamical Systems ◽  
Mixing Properties ◽  
Infinite Measure

scholarly journals Monomial dynamical systems of dimension one over finite fields

2011 ◽  
Vol 148 (4) ◽  
pp. 309-331 ◽  
Author(s):  
Min Sha ◽  
Su Hu
Keyword(s):  
Dynamical Systems ◽  
Finite Fields ◽  
Dimension One

scholarly journals Infinite-Volume Mixing for Dynamical Systems Preserving an Infinite Measure

2012 ◽  
Vol 5 ◽  
pp. 204-219 ◽  
Author(s):  
Marco Lenci
Keyword(s):  
Dynamical Systems ◽  
Infinite Volume ◽  
Infinite Measure

scholarly journals First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

2012 ◽  
Vol 194 (2) ◽  
pp. 793-830 ◽  
Author(s):  
Ian Melbourne ◽  
Dalia Terhesiu
Keyword(s):  
Dynamical Systems ◽  
Higher Order ◽  
Ergodic Theorems ◽  
Infinite Measure

scholarly journals Operator renewal theory and mixing rates for dynamical systems with infinite measure

2011 ◽  
Vol 189 (1) ◽  
pp. 61-110 ◽  
Author(s):  
Ian Melbourne ◽  
Dalia Terhesiu
Keyword(s):  
Dynamical Systems ◽  
Renewal Theory ◽  
Infinite Measure ◽  
Mixing Rates
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