Dynamics Done With Your Bare Hands: Lecture Notes by Diana Davis, Bryce Weaver, Roland K. W. Roeder, and Pablo Lessa

216 Pages·2016·6.057 MB·English

by  Francoise Dal’bo, Francois Ledrappier, Amie Wilkinson

Series of Lectures in Mathematics
Dynamics Done with Your Bare Hands D
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Françoise Dal’Bo, François Ledrappier and
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This book arose from 4 lectures given at the Undergraduate Summer School of the h

Thematic Program Dynamics and Boundaries held at the University of Notre Dame. It Y Your Bare Hands
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is intended to introduce (under)graduate students to the field of dynamical systems by u
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emphasizing elementary examples, exercises and bare hands constructions.
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The lecture of Diana Davis is devoted to billiard flows on polygons, a simple-sounding r Lecture notes by Diana Davis, Bryce Weaver,
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class of continuous time dynamical system for which many problems remain open. H
Roland K. W. Roeder, Pablo Lessa
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Bryce Weaver focuses on the dynamics of a 2 × 2 matrix acting on the flat torus. n
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This example introduced by Vladimir Arnold illustrates the wide class of uniformly s
hyperbolic dynamical systems, including the geodesic flow for negatively curved,
Françoise Dal’Bo
compact manifolds.
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Roland Roeder considers a dynamical system on the complex plane governed by a ienç
quadratic map with a complex parameter. These maps exhibit complicated dynamics Wo
related to the Mandelbrot set defined as the set of parameters for which the orbit ilkinsie D Amie Wilkinson
remains bounded. sonal’B
Pablo Lessa deals with a type of non-deterministic dynamical system: a simple walk on , Edo, F Editors
an infinite graph, obtained by starting at a vertex and choosing a random neighbor itra
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at each step. The central question concerns the recurrence property. When the graph rsço
is a Cayley graph of a group, the behavior of the walk is deeply related to algebraic is L
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ISBN 978-3-03719-168-2
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Dynamics Done with
Your Bare Hands
Lecture notes by Diana Davis, Bryce Weaver,
Roland K. W. Roeder, Pablo Lessa
Françoise Dal’Bo
François Ledrappier
Amie Wilkinson
Editors
Center for Mathematics
Summer School
Editors:
Prof. Françoise Dal’Bo Prof. Amie Wilkinson
IRMAR Department of Mathematics
Université de Rennes I University of Chicago
Campus de Beaulieu 5734 S. University Avenue
35042 Rennes Cedex Chicago, IL 60637
France USA
E-mail: [email protected] E-mail: [email protected]
Prof. François Ledrappier
Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556
USA
E-mail: [email protected]
2010 Mathematics Subject Classification: 37A, 37B, 37D,37F, 37H, 53A
Key words: Dynamical systems, geometry, ergodic theory, billards, complex dynamics, random walk, group
theory
ISBN 978-3-03719-168-2
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9 8 7 6 5 4 3 2 1
Preface
The theory of dynamical systems has its origins in mechanics. A basic motivating
problem, explored by the dynamical pioneers Henri Poincaré and George Birkhoff
around the turn of the 20th century, was to predict the motion of heavenly bodies.
Today,dynamicsisoneofthemostlivelyareasinmathematics,andthe“dynamical
approach” is used to solve problems in a range of mathematical areas, including
numbertheory,geometry,andanalysis. Togiveoneexample,dynamicaltechniques
haveledtosignificantrecentprogressontheLittlewoodconjectureonsimultaneous
Diophantineapproximations,whichdatesbacktothe1930s. Arecentbreakthrough
in three-dimensional topology, the proof of the surface subgroup conjecture, relies
in part on dynamical properties of geodesic flows. Some of these techniques are
touched on in these lectures, in particular Bryce Weaver’s exposition of Margulis’s
method. An application of complex dynamics to astrophysics appears in Roland
Roeder’slectures: asinglelightsourcecanhaveatmost5n−5imageswhenlensed
bynpointmasses. Andthelistgoeson.
Thismonographisintendedtointroducethereadertothefieldofdynamicalsys-
temsbyemphasizingelementaryexamples,exercises,andbare-handsconstructions.
These notes were written for the Undergraduate Summer School of the thematic
program“BoundariesandDynamics”heldin2015attheUniversityofNotreDame,
inpartnershipwiththeNSFandtheFrenchGDRPlaton3341CNRS.
Roughly speaking, a dynamical system is a space that can be transformed by a
fixed set of rules (classically these rules are deterministic, but in the last chapter,
randomdynamicsisexplored). Byapplyingtheserulesrepeatedly, underaprocess
called iteration, the space evolves over a discrete set of time intervals. In a slight
variation of this definition, the system evolves over a continuous time interval such
as the real numbers, still subject to the rules given by, for example, an ordinary
differential equation. In both settings, the object of the game is to understand the
future states of the system. Starting at a particular point in the space and following
itsfutureiterationsgivesanorbitortrajectoryofthesystem. Manyquestionsarise,
depending on the system. Are there bounded trajectories? Periodic orbits, which
return to their starting point after a finite period of time? Orbits that fill the space
densely? When do two systems have the same orbits in some sense, and what are
invariantsofasystemthatcandetectthistypeofequivalence? Cansuchinvariants
becomputedusingperiodictrajectories?
Billiard flows are a basic type of continuous time dynamical system and arise
naturallyasmodelsinphysics. Herethetableitselfgivesaframeworkforthespace,
andapointinthespace,imaginedasaballwithaspecifiedvelocity,travelsinthespace
overtimebyreflectingoffthesidesofthetable. Inthefirstchapter,DianaDavisstarts
withasquarebilliardtableandtakesustomoreexotictables,therationalpolygons
(i.e.,tableswhereallcorneranglesareequaltoafractionalpartofπ). Tablesgiverise
vi Preface
tosurfaces: unfoldingthebilliardtable,thetrajectoryoftheballcanbereimagined
as a “straight line” curve on a translation surface created from a polygon where
parallel edges are identified by translations. For the square, the associated surface
is a flat torus and resembles a donut. This particular surface is rich in symmetries,
like reflections across verticals and horizontals. A class of transformations of this
surfacecalledshearmapscanbeusedtounderstandthebehaviorofthestraightline
curves on the torus and hence to describe the trajectories on a square billiard table
(Chapter1,Theorem5.5). Periodictrajectoriesarecharacterizedbyaninitialvelocity
withrationalslope(Chapter1, Exercise2.1). Suchtrajectoriescanbegroupedinto
familiesofcylindersofparallelperiodictrajectorieswithequallengths. Analogous
techniques can be used to analyze rational billiards, with highly symmetric tables,
knownasVeechtables,admittingaparticularlycompleteunderstanding.
This deceptively simple-sounding class of dynamics continues to captivate re-
searchers,withmanyopenproblemsremaining.
For example, let N(r) be the number of cylinders in a rational billiard table
correspondingtoperiodictrajectoriesoflengthatmostr. Afamousconjecturestates
thatthelimitof N(r)/r2 existsandisnonzero. Thebestresulttodatewasgivenby
AlexEskin,MaryamMirzakhani(FieldsMedal2014!),andAmirMohammadi. The
beautyoftheirworkcomesfromthemethod: theydeduceasymptoticpropertiesof
thecountingfunctionN(r)fromthedescriptionoftrajectoriesofadynamicalsystem
defined on a very big space, the moduli space, in which the initial billiard is just a
point! For billiards on irrational polygons, few tools are available and not much is
known. Forexample,theexistenceofperiodictrajectoriesisanopenproblemeven
fortriangles.
Inthesecondchapter,BryceWeaverrestrictshisattentiontoadiscretedynamical
system on the flat torus defined by a 2×2 matrix A with integer coefficients and
determinant1. Theeigenvaluesofsuchamatrixaremultiplicativeinversesofeach
other;toavoidtrivialdynamics,weassumeoneoftheseeigenvaluesisrealandbigger
than1.
Thisexample,introducedinthe1960sbyVladimirArnold(andplayfullytermed
the “cat map”), is the quintessential model for the class of uniformly hyperbolic
dynamicalsystemsthatarehighlysensitivetoinitialconditions. Differentinvariants
expresstheunpredictablebehaviorofsuchsystems. Oneofthem,topologicalentropy,
measurestheexponentialrateatwhichpointsseparate. Fortheprocessgeneratedby
the matrix A, this invariant equals the logarithm of the biggest eigenvalue λ > 1 of
thematrix(Chapter2,Proposition3.9). Theentropyisconnectedtotheasymptotic
behavior of the counting function PO defined by the number of A-periodic points
n
of period less than n. Roughly speaking, this invariant corresponds to the growth
rate of PO. More precisely, the limit of PO × n/λn exists and equals λ/(λ − 1)
n n
(Chapter 2, Theorem 4.2). The chapter provides two proofs of this fundamental
relationship. The first one relies on the algebraic nature of the transformation A
Preface vii
and is elementary. The first proof even gives an explicit formula for the number of
A-periodic points of period n (Chapter 2, Theorem 4.5). The second proof uses an
argument based on the presence of expanding and contracting directions attached
to A (Chapter 2, Section 4.2) and on the fact that this transformation “mixes” the
sets (Chapter 2, Proposition 4.15). This approach, which combines geometry and
ergodic theory, was developed by Gregory Margulis. It is longer and less precise
thanthefirstone(Chapter2,Theorem4.5isreplacedbyaweakerversion,Theorem
4.11) but it applies to a vastly more general class of systems: those for which fine
algebraicinformation(suchastheeigenvaluesofamatrix)isnotavailable. Indeed,
Margulis’smethodcanbeusedtodeducegeometricalinformationaboutnegatively
curved compact manifolds, in particular the growth rate of the number of closed
geodesicsasafunctionoftheirlength.
Inthethirdchapter,RolandRoederconsidersadynamicalsystemonthecomplex
planeC,governedbyaquadraticmap p (z)= z2+c,wherecisacomplexnumber,
c
a parameter that can be changed to vary the dynamics of the system. The study
of this family of maps {p : c ∈ C} was initiated by two founders of holomorphic
c
dynamics, Pierre Fatou and Gaston Julia. This area came to explosive life in the
1980swiththeintroductionofso-calledquasiconformalmethodsonthetheoretical
sideand,ontheexperimentalside,withtheblossomingofthepersonalcomputeras
amathematicaltool.
Despitethesimplicityoftheirdefiningformula,themapsp exhibitcomplicated
c
dynamics. Points which are far from the origin O in C escape to infinity. Among
the other points, there is at most one periodic orbit around which spiral the orbits
ofnearbypoints. Thesetofparameters c forwhich p hasasingleattractingfixed
c
point is contained inside a cardioid (Chapter 3, Lemma 2.9). Increasing the size
(i.e.,period)oftheperiodicorbitattachesopenblobstothiscardioidinthecomplex
parameter plane. Collecting all of these blobs together, the set M0 of parameters c
for which the map p admits such attracting periodic points has a rich topological
c
and combinatorial structure. This set M0 is contained in the famous Mandelbrot
set M, defined as the set of parameters for which the orbit of O remains bounded.
(This set was named for Benoît Mandelbrot, who brought public attention to this
class of dynamical systems and its vivid computer images in the 1980s). A central
openquestionincomplexdynamicsaskswhentheclosureof M0 coincideswith M.
Althoughitisunsolved,itwasprovedin1997forathinsliceoftheparameterplane:
the restriction of M0 and M to the real line (i.e., for the parameters c being real
numbers). Theset M isstillmysterious,butmanyinterestingpropertiesareknown.
Inparticular, itisself-similar: M containsarbitrarilysmallcopiesofitself. Adrien
DouadyandJohnHubbardhaveprovedthatMisconnected,anditisconjecturedthat
M is locally connected. There is a deep relationship between M and the dynamics
ofanindividualmap p throughtheshapeofthefilledJuliaset K ofpointshaving
c c
a bounded orbit under p . Namely, K is connected if and only if c belongs to
c c
viii Preface
M (Chapter 3, Section 3). This principle is exploited to transfer information from
thedynamicsofanindividualmemberof M backtogeometricinformationabout M
itself. Inparticular,itisusedtoshowthattheboundaryofMhasamazingcomplexity:
unliketheboundaryofadisk,whichisasmoothone-dimensionalcurve,theboundary
ofM has(Hausdorff)dimension2. Toshowthelimitsofourunderstanding,itisnot
knownwhethertheboundaryof M mightevenhavepositivearea!
The last chapter deals with a type of nondeterministic dynamical system: a
random walk, for which the iteration at each step is governed by a probability law.
In a sense, this is a classical dynamical system in which the transformation rules
areallowedtoincludearollofthedice. Thetheoryofrandomwalks, whichmixes
geometry and probability, was first developed in the 1920s. It has incredibly broad
applicabilityandtoday,lessthanacenturylater,itisnearlyubiquitousinscienceand
engineering. Pablo Lessa concentrates on simple walks on a combinatorial object,
an infinite graph, obtained by starting at a vertex and choosing a random neighbor
ateachstep. Thecentralquestionconcernstherecurrenceproperty: doesthewalk
visit“almostsurely”everyvertexinfinitelymanytimes? Thesewalksareoneofthe
mostclassicalexamplesofhowthegeometryoftheunderlyingspaceinfluencesthe
behavior of stochastic processes on that space. The first result in this direction was
obtained by George Pólya for grids: the simple walk on the two-dimensional grid
Z2 isrecurrentbutonthethree-dimensionalgrid Z3 thewalkisnotrecurrent—itis
transient (Chapter 4, Section 2.7). The study of simple random walks on Zd is a
first step in understanding a more complicated object: a continuous time stochastic
process on Rd (or on Riemannian manifolds) called Brownian motion. The “wire
mesh” in Rd with vertices in the grid Zd is an example of a Cayley graph, which
encodesthestructureofafinitelygeneratedgroupG anditsgenerators(inthiscase
Zd with the standard generating set). For this special class of graphs, Nicholas
Varopoulosprovedthatrecurrenceoftherandomwalkcanbecharacterizedentirely
bycertainalgebraic/geometricpropertiesofthegroupG. Inparticular,ifwedefine
the counting function f (n) of G to be the number of words of length at most n
G
(relative to a set of generators), then the random walk on the Cayley graph of G is
recurrentifandonlyif f (n)isboundedabovebyapolynomialfunctionofdegreeat
G
most2(Chapter4,Section4). Thefieldofgeometricgrouptheorygrewinthe1980s
tostudytherelationshipbetweenthealgebraicpropertiesofgroupsandthegeometric
propertiesoftheirCayleygraphs. AtheoremprovedbyMikhailGromovillustrates
the deep relationship between these two objects: the counting function f (n) has
G
polynomialgrowthifandonlyifGadmitsafiniteindexnilpotentsubgroup.
FrançoiseDal’Bo September2015
FrançoisLedrappier
AmieWilkinson
Contents
1 Linesinpositivegenus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
DianaDavis
1 Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Thesquaretorus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Cuttingsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Revisitingbilliardsonthesquaretable . . . . . . . . . . . . . . . . 10
5 Symmetriesofthesquaretorus . . . . . . . . . . . . . . . . . . . . 13
6 Continuedfractions . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7 Continuedfractionsandcuttingsequences . . . . . . . . . . . . . . 20
8 Everyshearcanbeunderstoodviabasicshears . . . . . . . . . . . 24
9 Polygonidentificationsurfaces . . . . . . . . . . . . . . . . . . . . 28
10 VerticesandtheEulercharacteristic . . . . . . . . . . . . . . . . . 31
11 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
12 Square-tiledsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 40
13 Regularpolygonsandthemodulusmiracle . . . . . . . . . . . . . . 43
14 Billiardsontriangulartables . . . . . . . . . . . . . . . . . . . . . 47
15 Wardsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
16 Bouw–Möllersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 49
17 Teichmüllerspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 Introductiontocomplicatedbehaviorandperiodicorbits . . . . . . . . . . 57
BryceWeaver
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Definitionsandfirstexamples . . . . . . . . . . . . . . . . . . . . . 59
3 Complicatedsystemsandstructures . . . . . . . . . . . . . . . . . 71
4 Countingperiodicorbits . . . . . . . . . . . . . . . . . . . . . . . 82
5 Finalremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6 Appendix: Proofofmixing . . . . . . . . . . . . . . . . . . . . . . 95
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3 Aroundtheboundaryofcomplexdynamics. . . . . . . . . . . . . . . . . 101
RolandK.W.Roeder
1 Warmup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2 Mandelbrotsetfromtheinsideout . . . . . . . . . . . . . . . . . . 120
3 Complexdynamicsfromtheoutsidein . . . . . . . . . . . . . . . . 133
4 Complexdynamicsandastrophysics . . . . . . . . . . . . . . . . . 145
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153