Table Of ContentErgebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge
A Series of Modern Surveys in Mathematics 68
Viviane Baladi
Dynamical
Zeta Functions
and Dynamical
Determinants for
Hyperbolic Maps
A Functional Approach
Ergebnisse der Mathematik und Volume 68
ihrer Grenzgebiete
3. Folge
A Series of Modern Surveys
in Mathematics
EditorialBoard
L.Ambrosio,Pisa V.Baladi,Paris
G.-M.Greuel,Kaiserslautern
M.Gromov,Bures-sur-Yvette
G.Huisken,Tu¨bingen J.Jost,Leipzig
J.Kolla´r,Princeton G.Laumon,Orsay
U.Tillmann,Oxford J.Tits,Paris
D.B.Zagier,Bonn
Forfurthervolumes:
www.springer.com/series/728
Viviane Baladi
Dynamical Zeta Functions
and Dynamical Determinants
for Hyperbolic Maps
A Functional Approach
VivianeBaladi
IMJ-PRG
CNRSandSorbonneUniversite´
Paris,France
ISSN0071-1136 ISSN2197-5655(electronic)
ErgebnissederMathematikundihrerGrenzgebiete.3.Folge/ASeriesofModernSurveys
inMathematics
ISBN978-3-319-77660-6 ISBN978-3-319-77661-3(eBook)
DOI10.1007/978-3-319-77661-3
LibraryofCongressControlNumber:2018943138
MathematicsSubjectClassification:37C30,37D20,37D35
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Preface
On July 9, 2001, Springer invited me to contribute a monograph on dynam-
ical systems. I immediately accepted, offering to write a book on dynami-
cal zeta functions and dynamical determinants: Dynamical determinants are
functions defined from weighted periodic orbit data of a differentiable dy-
namical system. The zeroes of these functions describe a large part of the
spectrum of an associated transfer operator. In other words they play the
role of Fredholm determinants for the (usually not compact) transfer opera-
tor. This spectrum contains key information on the statistical properties of
the dynamics. Dynamical zeta functions1 can often be written as an alter-
nating product of dynamical determinants, and their poles hence describe a
usually smaller part of the spectrum of the transfer operator. Deep impor-
tant results on dynamical zeta functions and dynamical determinants were
obtained in the 80s and 90s by Ruelle and Pollicott, and the corresponding
spectra are often called Ruelle–Pollicott resonances.
In2001,inspiteoftheexistenceinpreprintformofthepioneeringworkof
Blank–Keller–Liverani[37],Icouldnotforeseethemajorchangesthatwould
occur in the theory in the next decade. What Blank–Keller–Liverani did, for
an Anosov diffeomorphism T :M →M, was to construct a Banach space of
anisotropic distributions on M on which the transfer operator
ϕ◦T−1
Lϕ= , (0.1)
|detDT ◦T−1|
(defined initiallyforϕ∈L∞(M)) had aspectral gap. This wasthe first time
that a spectral gap had been obtained for a transfer operator of a hyperbolic
mapwithoutusingsymbolicdynamics.SinceMarkovpartitionsandthepas-
sagefromaninvertibletoanexpanding(one-sided)symbolicdynamicscause
a great loss of information (starting from a Cr diffeomorphism, one is re-
1 The 1990 bookby Parry andPollicott [131] is the primaryreference on dynamical
zetafunctionsforhyperbolicdynamics,butitdoesnotcoverdynamicaldeterminants,
the symbolicdynamicsapproachusedthere beingunsuitableforthis purpose.
v
vi Preface
duced to a one-sided Ho¨lder shift), the work of Blank–Keller–Liverani was
an important stepping-stone in the theory of smooth chaotic dynamics. We
call this symbolic dynamics-free spectral approach, using anisotropic spaces
of distributions, the functional approach.
Between2005and2008,sevenpapers2byvariousauthors(Liverani,Tsujii,
Gou¨ezel, myself) appeared ([15, 119, 120, 87, 88, 28, 31]), generalising and
enhancing the results of Blank–Keller–Liverani, and connecting the spectral
data to zeroes of dynamical determinants
(cid:3)
(cid:2)∞ zm (cid:2) m−1g(Tk(x))
dT−1,g(z)=exp− m |det(Ikd=0−DT−m(x))|,
m=1 Tm(x)=x
for hyperbolic diffeomorphisms T, and g = |detDT−1|, or more general
weights.
TherearetwobasickindsofanisotropicBanachspaces(see[17]):Liverani
and co-workers use a geometric definition [119, 120, 87, 88], based on cones
in the tangent space. The approach of [28, 31] uses cones in the co-tangent
spaceviatheFouriertransform,inthespiritofSobolevspaces,anditcanbe
called microlocal. This microlocal approach has been adopted and developed
by the semi-classical community, defining anisotropic Hilbert spaces for C∞
dynamics.(Weonlymentionherethediscrete-timeresultsofFaure–Roy[67],
Faure–Roy–Sjo¨strand[68],referringtoGou¨ezel[86]andZworski[184,§4]and
references therein for flows.)
Over the past few years, a clear picture of the functional approach has
gradually emerged. The time seemed finally ripe for a book. So I was de-
lighted when Ute Motz contacted me again in 2009 to nudge me gently into
getting my act together. A substantial amount of time in the preparation of
this book has then been spent over agonising decisions on what to include:
I have decided to leave out completely both continuous-time dynamics and
piecewise smooth dynamics. The goal of this book, intended for researchers
and graduate students in dynamical systems, is thus to give a self-contained
and (hopefully) reader-friendly account of the “microlocal” version [28, 31]
of the functional approach, in the setting of finitely differentiable hyperbolic
diffeomorphisms. In order to allow easier entry into the topic, we present
the arguments first in the much easier toy model of differentiable (locally)
expandingmaps.Oneofthefeaturesofthisbookisthatnoknowledge3 ofmi-
crolocal analysis or pseudodifferential operator theory is requested. We hope
that this will demystify the construction and the use of anisotropic spaces,
as well as demonstrate that pedestrian (dyadic decomposition) techniques in
Fourier space give very precise results. The book also aims to show the role
2 A previous important paper by Kitaev [112] was devoted to the dynamical deter-
minant,with nospectral interpretation.
3 SeeSection 1.4forthe very shortlist ofblack boxesthat we use.
Preface vii
played by the kneading determinant ideas of Milnor and Thurston [123] in
this theory.
I would like to express here my deep thanks to David Ruelle, Ger-
hard Keller, Predrag Cvitanovi¸c, Mark Pollicott, Hans Henrik Rugh, David
Fried,CarlangeloLiverani,S´ebastienGou¨ezel,MasatoTsujii,Fr´ed´ericNaud,
St´ephane Nonnenmacher, Maciej Zworski, and Semyon Dyatlov (in chrono-
logical order) for many enlightening conversations over the years. I thank
Dmitry Todorov, for questions on Sections 2.2–2.4 of a preliminary draft
and Alexander Adam who found several typos in preliminary versions. I am
much indebted to Malo J´ez´equel for a very sharp critical reading of the
September 2016 version. I also thank the four reviewers of the September
2016 manuscript for constructive remarks.
I am also grateful to the audiences of several mini-courses for their
questions: In Rio de Janeiro (IMPA) in 2007, in Vienna in 2008 (Erwin
Schr¨odinger Institute, semester on Hyperbolic Dynamics), in Porto in 2010,
in Paris (UPMC) and Pisa in 2016. A short stay at the Newton Institute
in the Fall of 2013 helped me go back to the book project after two diffi-
cult years. The most crucial phases of the writing were achieved during the
Summers of 2010, 2011, 2014, 2015, 2016, and 2017 in the Centre for Mathe-
maticalSciences,LundUniversity,whichprovidedanoffice—andfreeaccess
to the printer for several thousands of pages. I am also very grateful to the
Mittag-Leffler Institute in Djursholm, where the book was finally completed
in the Fall of 2017 during the Research Program on Fractal Geometry and
Dynamics.
Last, but not least, I thank Tomas Persson, for reading and commenting
on several drafts of this book, and much more.
Djursholm, December 3, 2017 Viviane Baladi
Contents
1 Introduction.............................................. 1
1.1 Statistical properties of chaotic differentiable dynamical
systems ............................................... 1
1.2 Transfer operators. Dynamical determinants. Resonances .... 6
1.3 Main results. Examples ................................. 10
1.4 Main techniques........................................ 14
Comments ................................................. 16
Part I Smooth expanding maps
2 Smooth expanding maps: The spectrum of the transfer
operator .................................................. 21
2.1 Transfer operators for smooth expanding maps on H¨older
functions .............................................. 21
2.2 Transfer operators for smooth expanding maps on Sobolev
spaces................................................. 28
2.2.1 Isotropic Sobolev spaces Ht and good systems of
p
charts........................................... 29
2.2.2 Bounding the essential spectral radius (Theorem 2.15) 32
2.2.3 The key local Lasota–Yorke bound (Lemma 2.21)..... 36
2.2.4 Fragmentation and reconstitution: Technical lemmas.. 38
2.3 The essential spectral radius on Sobolev spaces: Interpolation 41
2.3.1 Complex interpolation ............................ 42
2.3.2 Proof of Theorem 2.15 on Ht for integer
p
differentiability................................... 44
2.4 The essential spectral radius: Dyadic decomposition......... 48
2.4.1 A Paley–Littlewood description of Ht and Ct ........ 50
p ∗
2.4.2 Proof of Lemma 2.21 and Theorem 2.15: The general
case ............................................ 53
ix
x Contents
2.5 Spectral stability and linear response `a la Gou¨ezel–Keller–
Liverani............................................... 64
Problems .................................................. 74
Comments ................................................. 76
3 Smooth expanding maps: Dynamical determinants........ 79
3.1 Ruelle’s theorem on the dynamical determinant ............ 79
3.1.1 Dynamical zeta functions.......................... 83
3.2 Ruelle’s theorem via kneading determinants................ 86
3.2.1 Outline ......................................... 86
3.2.2 Flat traces ...................................... 88
3.3 Dynamical determinants: Completing the proof of
Theorem 3.5 ........................................... 93
3.3.1 Proof of Theorem 3.5 if α>d+t .................. 94
3.3.2 Nuclear power decomposition via approximation
numbers ........................................ 99
3.3.3 Asymptotic vanishing of flat traces of the
non-compact term ................................ 104
3.3.4 The case α≤d+t of low differentiability ........... 111
Problems .................................................. 116
Comments ................................................. 118
Part II Smooth hyperbolic maps
4 Anisotropic Banach spaces defined via cones.............. 123
4.1 Transfer operators for hyperbolic dynamics ................ 123
4.1.1 Hyperbolic dynamics and anisotropic spaces ......... 123
4.1.2 Bounding the essential spectral radius (Theorem 4.6) . 128
4.1.3 Reducing to the transitive case..................... 130
4.2 The spaces Wpt,,∗s and Wpt,,∗s∗ .............................. 132
4.2.1 Charts and cone systems adapted to (T,V) .......... 132
4.2.2 Formal definition of the spaces Wpt,,∗s and Wpt,,∗s∗....... 134
4.3 The local Lasota–Yorke lemma and the proof of Theorem 4.6 137
4.3.1 The Paley–Littlewood description of the spaces and
the local Lasota–Yorke lemma ..................... 138
4.3.2 Fragmentation, reconstitution, and the proof of
Theorem 4.6 ..................................... 148
Problems .................................................. 152
Comments ................................................. 153
5 A variational formula for the essential spectral radius..... 157
5.1 Yet another anisotropic Banach space: Bt,s ................ 157
5.1.1 Defining Bt,s..................................... 162
5.2 Bounding the essential spectral radius on Bt,s (Theorem 5.1). 165
5.3 Spectral stability and linear response...................... 173
Contents xi
Problems .................................................. 181
Comments ................................................. 182
6 Dynamical determinants for smooth hyperbolic dynamics. 183
6.1 Dynamical determinants via regularised determinants and
flat traces ............................................. 183
6.2 Proof of Theorem 6.2 on d (z) if r−1>d+t−s......... 187
T,g
6.3 Theorem 6.2 in low differentiability r−1≤d+t−s........ 196
6.4 Operators on vector bundles and dynamical zeta functions... 200
Problems .................................................. 206
Comments ................................................. 207
7 Two applications of anisotropic spaces .................... 209
7.1 Equilibrium measures and SRB measures.................. 209
7.1.1 Peripheral spectrum and equilibrium measures ....... 210
7.1.2 Grassmannians and the measure of maximal entropy.. 223
7.1.3 SRB measures for hyperbolic attractors ............. 225
7.2 Tsujii’s proof of Anosov’s theorem ........................ 230
Problems .................................................. 232
Comments ................................................. 234
Part III Appendices
A Spectral theory ........................................... 237
A.1 Bounding the essential spectral radius: Hennion’s theorem ... 237
A.2 Eigenvalues and eigenvectors for different Banach spaces..... 238
A.3 An abstract perturbation result of Gou¨ezel–Keller–Liverani .. 240
A.4 Nuclear operators and approximation numbers ............. 245
B Thermodynamic formalism: Non-multiplicative
topological pressure ...................................... 249
C Properly supported operators (pseudolocality)............ 259
D Alternative proofs for C∞ dynamics and weights ......... 263
D.1 Elements of symbolic calculus ............................ 263
D.2 Essential spectral radius for C∞ expanding maps........... 269
D.3 Dynamical determinants for C∞ expanding maps........... 271
D.4 The essential spectral radius for C∞ hyperbolic maps ....... 275
References.................................................... 281
Index......................................................... 289
