Project options
Statistical properties in hyperbolic
dynamics
S1.
Rufus Bowen, Markov
partitions for Axiom A diffeomorphisms, Amer. J. Math. 92
(1970), p. 725-747.
This paper proves that uniformly hyperbolic systems can be modeled with
subshifts of finite type.
S1++
Bowen, Rufus.
Equilibrium states and the ergodic theory of
Anosov diffeomorphisms, Chapter 3.
Second revised edition. With a preface by David Ruelle. Edited by
Jean-René Chazottes. Lecture Notes in Mathematics, 470.
Springer-Verlag, Berlin, 2008. viii+75 pp.
Compared
to S1, this is more readable and
uses a different approach (more transparent and indeed better). This,
and not the approach from Bowen's 1970 paper, is the approach that
Sarig generalizes for surface diffeos.
S2.
C.
Liverani, Central limit theorem for
deterministic systems, in International Conference on
Dynamical Systems (Montevideo, 1995), 56-75, Pitman Res. Notes Math.
Ser., 362, Longman, Harlow, 1996.
preprint
on Liverani's page
Once it has been proved that correlations decay quickly
enough, the CLT can be derived using martingale
approximations.
S3.
Carlangelo
Liverani, Decay of correlations in
piecewise expanding maps, Journal of Statistical
Physics 78 (1995), p. 1111-1129.
Carlangelo Liverani, Benoit Saussol, and Sandro
Vaienti. A probabilistic approach to
intermittency. Ergodic Theory Dynam. Systems 19 (1999),
no. 3, 671-685.
The first paper carries out the details of the proof of decay
of correlations using the method of cones and the Hilbert
metric, which might be discussed in Will Ott's lectures.
The second extends this to some non-uniformly hyperbolic
dynamical systems, which display "intermittent" chaotic
behavior. NOTE that although a particular cone plays a central
role, this is not related to the Hilbert metric.
S4.
Gerhard Keller and Carlangelo Liverani:
A spectral gap for a one-dimensional lattice of
coupled piecewise expanding interval maps, Section 2,
Lect. Notes Phys. 671 (2005), p. 115-151
Self-contained proof - with background on BV - of exponential decay of
correlations for piecewise-expanding maps on the interval, including
the Lasota-Yorke inequality.
S5.
Ian Melbourne and Matthew Nicol:
Almost sure invariance principle for nonuniformly hyperbolic
systems, Sections 2 (a)-(b),
Commun. Math. Phys. 260 (2005) 131-146.
A clean discussion of exponential decay of correlations for
Gibbs-Markov maps (includes full-branch piecewise expanding interval
maps)
Statistical mechanics and thermodynamic
formalism
T1
Bowen, Rufus.
Equilibrium states and the ergodic theory of
Anosov diffeomorphisms, Chapter 1.
Second revised edition. With a preface by David Ruelle. Edited by
Jean-René Chazottes. Lecture Notes in Mathematics, 470.
Springer-Verlag, Berlin, 2008. viii+75 pp.
Develops the general theory of Gibbs
measures for Holder continuous potential functions on topological
Markov chains, starting with the motivation from thermodynamics.
T2
Bowen, Rufus.
Some systems with unique equilibrium
states.
Math. Systems Theory 8 (1974/75), no. 3, 193-202.
Gives an approach to existence and
uniqueness of equilibrium states that does not use any symbolic coding
of the system, but rather relies on the "specification" property, which
stipulates that an arbitrary collection of finite-length trajectories
of the system can be shadowed by a single trajectory that takes a
uniformly bounded time to transition from one orbit segment to the
next.
T3
D.V. Anosov, A.V. Klimenko, G. Kolutsky.
On the hyperbolic automorphisms of the 2-torus
and their Markov partitions,
(2008),
arXiv:0810.5269
Starts with a discussion of how to
treat deterministic systems as random processes, then constructs Markov
partitions on the 2-torus for hyperbolic automorphisms, which lets
these systems be viewed as Markov shifts. This was the first example of
Markov partitions historically and is simpler than the general
construction by Bowen in S1. The
paper also includes some results on classification of Markov
partitions.
T4
Lai-Sang Young.
What are SRB measures, and which dynamical
systems have them?
J. Stat Phys 108, No 5/6 (2002), 733-754.
The use of mathematical methods from
statistical mechanics to construct a physically relevant SRB measure
for smooth hyperbolic dynamical systems is one of the crowning
achievements of thermodynamic formalism. This survey paper gives an
overview.
Dynamics of quantum spin systems
All papers and books given here are open-access and are available online.
If the arxiv reference for a paper or book is not given, use
scholar.google.com to search for it.
Listed in the difficulty order. First
three are readily accessible.
Q1-Q2
(Uncertainty principle)
Box 2.4 (p.89) in Michael Nielsen and Isaac
Chuang, Quantum computation and quantum
information, Cambridge University Press, 2010
(Bell inequality)
Chapter 2.6 in M. Nielsen and I. Chuang,
Quantum computation and quantum
information, Cambridge University Press, 2010
Q3
(Quantum teleportation protocol)
Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard
Jozsa, Asher Peres, and William K. Wootters.
Teleporting an unknown quantum state via dual
classical and Einstein-Podolsky-Rosen channels, Physical Review
Letters, 70(13):1895-1899, (1993)
Alternatively (easier to read): Chapter 6.2 in Mark
Wilde, From Classical to Quantum Shannon
Theory, Cambridge University Press, 2013 arXiv:1106.1445
Q4
(Short review of Lieb-Robinson bounds
and applications)
B. Nachtergaele, R. Sims, Much Ado About
Something: Why Lieb-Robinson bounds are useful, IAMP News
Bulletin, October 2010, 22-29, (2010) arXiv:1102.0835
Q5
(Paper on which these lectures are
based)
B. Nachtergaele, Y. Ogata, R. Sims, Propagation of
Correlations in Quantum Lattice Systems, J. Stat. Phys. 124, 1-13,
(2006) arXiv:0603064
Q6
(Improvement of the constant in
Lieb-Robinson bound+applications)
B. Nachtergaele, R. Sims. Locality Estimates
for Quantum Spin Systems, Sidoravicius, Vladas (Ed.), New Trends
in Mathematical Physics. Selected contributions of the XVth
International Congress on Mathematical Physics, Springer Verlag,
591-614, (2009) arXiv:0712.3318
Q7
(Addition of on-site local
perturbations to the Hamiltonian in Lieb-Robinson bound+more
complicated systems)
B. Nachtergaele, H. Raz, B. Schlein, R.
Sims, Lieb-Robinson bounds for harmonic and
anharmonic lattice systems, Commun. Math. Phys. 286, 1073-1098,
(2009) arXiv:0712.3820
Q8
(Original Lieb-Robinson bounds paper
with different proof)
E.H. Lieb, D.W. Robinson, The finite group
velocity of quantum spin systems, Commun. Math. Phys. 28 (1972),
251-257.
Dynamical approaches to the spectral theory of
operators
The first two items are survey papers;
the other two contain results that will be mentioned in the
talks.
OP1
D. Damanik,
Schrödinger operators with dynamically
defined potentials,
Ergodic Theory Dynam. Systems 37 (2017), 1681-1764
(https://arxiv.org/abs/1410.2445)
OP2
S. Jitomirskaya,
Ergodic Schrödinger operators (on one
foot),
Spectral theory and mathematical physics: a Festschrift in honor of Barry
Simon's 60th birthday, 613-647, Proc. Sympos. Pure Math., 76, Part 2,
Amer. Math. Soc., Providence, RI, 2007
OP3
D. Damanik, A. Gorodetski, W. Yessen,
The Fibonacci Hamiltonian,
Invent. Math. 206 (2016), 629-692
(https://arxiv.org/abs/1403.7823)
OP4
A. Avila, J. Bochi, D. Damanik,
Cantor spectrum for Schrödinger operators
with potentials arising from generalized skew-shifts ,
Duke Math. J. 146 (2009), 253-280
(https://arxiv.org/abs/0709.2667)
Dynamics on homogeneous spaces, with applications to
number theory
H1
(a paper explaining Ratner's theorem in
the simple setting of SL(2,R), in a concrete way)
Manfred Einsiedler,
Ratner's theorem on
SL(2,R)-invariant measures
H2
(a paper giving an upper bound in a
quantitative Oppenheim conjecture)
Eskin, Margulis,
Mozes, Upper
Bounds and Asymptotics in a Quantitative Version of the Oppenheim
Conjecture
H3
(this paper establishes a quantitative
version of Oppenheim's conjecture for generic ternary indefinite
quadratic forms using an analytic number theory approach; the
statements come with power gains and in some cases are essentially
optimal)
Jean Bourgain,
A
quantitative Oppenheim Theorem for generic diagonal quadratic
forms
H4
(a survey paper on homogeneous dynamics
and number theory based on his talk at ICM 2010)
Elon
Lindenstrauss,
Equidistribution
in homogeneous spaces and number theory, in Proceedings of
the International Congress of Mathematicians, Hyderabad 2010
Funding for this event is provided by the NSF grant DMS-1800669.
