Ergodic Theory, Open Dynamics, and Coherent Structures

240 Pages·2014·4.258 MB·English

by  Wael Bahsoun, Christopher Bose, Gary Froyland (eds.)

Springer Proceedings in Mathematics & Statistics
Wael Bahsoun
Christopher Bose
Gary Froyland Editors
Ergodic Theory,
Open Dynamics,
and Coherent
Structures
Springer Proceedings in Mathematics & Statistics
Volume 70
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Wael Bahsoun • Christopher Bose • Gary Froyland
Editors
Ergodic Theory, Open
Dynamics, and Coherent
Structures
123
Editors
WaelBahsoun ChristopherBose
DepartmentofMathematicalSciences DepartmentofMathematicsandStatistics
LoughboroughUniversity UniversityofVictoria
Leicestershire,UK Victoria,BC,Canada
GaryFroyland
SchoolofMathematicsandStatistics
UniversityofNewSouthWales
Sydney,Australia
ISSN2194-1009 ISSN2194-1017(electronic)
ISBN978-1-4939-0418-1 ISBN978-1-4939-0419-8(eBook)
DOI10.1007/978-1-4939-0419-8
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Preface
Ergodic theory, as a mathematical discipline, refers to the analysis of asymptotic
or long-range behaviour of a dynamical system, that is, a map or flow on a
state space, using measure-theoretic or probabilistic methods. A close cousin to
smooth dynamics (the study of differentiable actions on a smooth manifold) and
to topological dynamics (comprising a continuous action on a topological space),
thereisawell-establishedandrichsynergybetweenthethreefields.Indeed,many
importantapplicationsbringtoolsfromallthethreefieldstobearinthestudyofa
particulardynamicalsystem.
As a quickly maturing mathematical field, both theoretical developments and
applications inthephysicalsciences,engineering, andcomputer scienceareflour-
ishing within the arena of modern research in ergodic theory. Driven by these
newtheoreticaltoolsandagrowingbreadthofnaturalapplications,computational
aspectsarenowacentralchallengetoresearchersinthefield.
An open dynamical system is a natural extension of the traditional (closed)
dynamical system. In an open system, the state space is no longer deemed to
be invariant under the dynamical action, but some orbits are allowed to ‘escape’
dependingonlocationandtime.Aneverydayexampleisthedynamicsofaballon
a billiard table; when the ball falls in a hole in the table, the orbit is terminated.
AsintroducedbyYorkeandPianigianiinthelate1970s,theabstractconceptofan
open system leads immediately to the notion of a conditionally invariant measure
andescaperatealongwithahostofdetailedquestionsabouthowmassescapesor
failstoescapefromthesystemundertimeevolution.
Perhaps ironically, concepts from open systems have recently been used to
analysetraditional,closedsystems.Forexample,inmanyclosedsystems,relaxation
to equilibrium is by no means uniform throughout the state space. There may be
regions that remain ‘almost invariant’ for long periods of time, mixing with the
rest of the space at quantifiably slower rates than the other parts of the system.
These ‘almost invariant sets’ become key features determining the asymptotics
of the system. One particularly fruitful idea is to study almost invariant sets as
open subsystems of the larger closed dynamical system, wherein the escape rate
determinestherateofmixingandrelaxationtoequilibrium.
v
vi Preface
In some realistic applications, time-varying parameters governing the flow or
transformation on the state space necessitate modelling by a non-autonomous
system. While the ergodic theory of non-autonomous systems parallels that of
autonomous dynamics in many ways, there are important differences. Stable and
unstablefoliations,afoundationofgeometricanalysisforanautonomousmapora
flow,becomeequivariant,time-dependentstructures.Otherdynamicalobjectssuch
as Lyapunov exponents and Oseledets subspaces can be used in alternative ways
to describe non-autonomous or random dynamics. Invariant or almost invariant
objects arising in autonomous dynamics have non-autonomous analogues called
coherent structures. These are features that move around in the state space under
time evolution but that may still represent barriers to mixing and relaxation
to ‘equilibrium’, a concept that also has to be reinterpreted compared to the
autonomoussetting.
From April 9 through April 15, 2012 a group of more than 40 researchers
in ergodic theory gathered at the Banff International Research Station in Banff,
Alberta, Canada to exchange cutting-edge developments in the field.1 Thirty-five
research talks were given during the course of the workshop, covering theoretical,
applied,andcomputationalaspectsofbothopenandclosed,autonomousandnon-
autonomous dynamics. After the workshop, a number of participants volunteered
to expand on their presentations and contribute chapters to this volume. Each
contribution was rigorously peer-reviewed before inclusion in this volume. We
brieflyoutlinetheresultingcontributions:
• Balasuriya considers time-dependent flows where the time dependence enters
as a perturbation of an autonomous flow. He describes how to analytically
estimate the perturbed stable and unstable manifolds, which may be regarded
asLagrangiancoherentstructures.HethenusesMelnikovtheorytoquantifyflux
acrosstheseperturbedmanifolds.
• Bandtlow and Jenkinson consider the spectrum of transfer operators of real-
analytic expanding maps acting on holomorphic functions of the interval and
other finite-dimensional spaces. They particularly consider the open setting
wheremassisleavingthephasespaceandproveboundsforeachspectralpoint
ofthecorrespondingopenoperators.
• Bandtlow,Jenkinson,andPollicottspecialisethepreviouschaptertothesetting
of piecewise real-analytic expanding Markov maps of the interval, with escape
throughaMarkovhole.Theyshowthattheleadingspectralpointofthetransfer
operator,whichquantifiestheescaperate,canbeapproximatedusingderivative
informationfromallperiodicpointsofincreasingperiod.Theinvariantmeasure
onthesurvivorsetisalsoestimated.
• BasnayakeandBolltdescribeamethodofextractingaflowfieldfromamovieof
observations.Undertheassumptionofsmoothtimedependenceoftheflowfield,
1Materials from this workshop, including abstracts and some videos of the presentations, are
availableattheBIRSwebsite,www.birs.math.ca;searchonworkshopcode12w5050.
Preface vii
theyintroduceamulti-stepmethodtoenforcesmoothbehaviour.Asanexample,
they extract a flow field from a movie of sea surface temperature and calculate
Lagrangian coherent structures in the form of finite-time Lyapunov exponent
fields.
• BoseandMurrayextendtheirearlierworkonestimatingabsolutelycontinuous
invariant measures (ACIMs) to the open dynamics setting. In general, an open
system may support a continuum of escape rates and an infinity of absolutely
continuousconditionalinvariantmeasures(ACCIMs).Theirapproach,basedon
maximumentropyandconvexoptimisation,allowsonetoprescribethedesired
escaperateandfindthecorrespondingACCIM.
• BruinstudiesamaponaEuclidean.d(cid:2)1/-dimensionaltrianglethatarisesfrom
a simple d-dimensional subtractive algorithm. For d D 2 the map becomes
the well-known Farey map on Œ0;1(cid:2); for d D 3, Bruin shows that the map
is dissipative but at the same time ergodic with respect to two-dimensional
Lebesguemeasureandevenexact.Thispapercontributestoalongandhistori-
callyimportantdevelopmentofnumbertheoreticapplicationsofergodictheory
dating back to Renyi in the 1950s with his foundational work on the continued
fraction expansion. At the same time, the tools used are up to date, bringing
modernnotionssuchasdistortion,Schwarzianderivative,andrandomwalksto
bearontheproblem.
• Bunimovich and Webb consider piecewise differentiable expanding Markov
mapsoftheintervalwithaMarkovhole.Theirfocusisonestimatingthesurvival
and escape probabilities after a finite number of iterations of the open system.
Theyprovideexplicitupperandlowerboundsfortheseprobabilitiesintermsof
eigenvaluesofthetransitionmatricesofinducedMarkovchains.
• Demersstudiesbilliarddynamicalsystemswithavarietyofholes.Usingtransfer
operatorandYoungtowertechniques,heprovestheexistenceofanaturalescape
rate and corresponding absolutely continuous conditional invariant measure
(ACCIM).Hethenconsidersthequestionofstabilityasthesizeoftheholegoes
tozeroandshowsthelimitingACCIMistheSRB(orphysical)measureforthe
closedsystem.Finallyheshowsthattheescaperatealsoarisesviaavariational
principle.
• Froyland and Padberg-Gehle give an overview of transfer operator methods
for finite-time almost-invariant and coherent sets. Their chapter unifies the
autonomous and time-dependent methodologies and then focuses on three
aspects, namely the flow direction, the flow duration, and the level of diffusion
present.Theyshowthatthecoherentstructuresproducedbythetransport-based
transferoperatorapproachareverynaturalfromageometricdynamicalpointof
view.
• Haydn,Winterberg,andZweimüllerconsiderageneralergodicprocessandthe
return time and hitting time distributions corresponding to a sequence of sets
of decreasing size. They first show that as the size of the sets approaches zero,
limitingreturnandhittingtimedistributionsexist.Further,theyshowthatifone
induces the original ergodic process via return times to a fixed set of positive
viii Preface
measure,thelimitingdistributionsofthereturnandhittingtimesoftheoriginal
andinducedsystemscoincide.
The contributions to this book represent a broad cross section of the topics
represented at the April 2012 workshop and, in turn, make a fine collection of
sample papers for researchers who may be looking to broaden their outlook in
modernaspectsofthefield.Theeditorswishtothankalltheworkshopparticipants
fortheircontributions,butespeciallythoseparticipantswhotookthetimetowrite
up their work as a submission to this book and the referees who helped to hone
theauthor’scontributionsintothehigh-qualityresearchpapersyouwillfindinthe
followingpages.
Finally, none of this would have been possible without the remarkable support
offeredbytheBanffInternationalResearchStationanditsstaff.BIRSisindeedone
of only a handful of first-class mathematical research venues in the world; if you
haveachancetogothere,donothesitate!
Leicestershire,UK WaelBahsoun
Victoria,BC,Canada ChristopherBose
Sydney,Australia GaryFroyland
July2013
Contents
1 Nonautonomous Flows as Open Dynamical Systems:
CharacterisingEscapeRatesandTime-VaryingBoundaries ......... 1
SanjeevaBalasuriya
1.1 Introduction........................................................... 1
1.2 ThePerturbativeSetting ............................................. 6
1.3 Boundaries Between Open Dynamical Systems:
InvariantManifolds .................................................. 9
1.4 FluxQuantification................................................... 13
1.5 SimplificationsofFluxFormulæintheSubcases .................. 20
1.6 ConcludingRemarks................................................. 24
References.................................................................... 25
2 EigenvaluesofTransferOperatorsforDynamicalSystems
withHoles ................................................................... 31
OscarF.BandtlowandOliverJenkinson
2.1 Introduction........................................................... 31
2.2 EigenvalueEstimatesviaWeyl’sInequality ........................ 33
2.3 TheCommonSpectrum.............................................. 36
2.4 HilbertHardySpace.................................................. 37
References.................................................................... 38
3 PeriodicPoints,EscapeRatesandEscapeMeasures................... 41
OscarF.Bandtlow,OliverJenkinson,andMarkPollicott
3.1 Introduction........................................................... 41
3.2 TransferOperatorsandDeterminants............................... 43
3.3 DeterminingtheEscapeRate........................................ 47
3.4 DeterminingtheEscapeMeasure.................................... 48
3.5 AnExample .......................................................... 51
3.5.1 TheEscapeRate............................................ 53
3.5.2 TheEscapeMeasure........................................ 55
References.................................................................... 57
ix