**2016
UH Summer school in dynamics: Project
options**

**Hyperbolic
dynamics and beyond**

Michael Shub.

Endomorphisms
of compact differentiable manifolds. Amer.
J. Math. 91 1969 175-199.

*Global
structural stability of expanding maps on
manifolds. Also some generic theory is
discussed.*

Anthony Manning.

There
are no new Anosov diffeomorphisms on tori.
Amer. J. Math. 96 (1974)

*Shows that the
matrix associated to an Anosov
diffeomorphism of the torus is a hyperbolic
matrix.*

Boris Kalinin.

Livsic theorem for matrix cocycles. Ann.
of Math. (2) 173 (2011), no. 2, 1025-1042.

*The title says
it all.*

Albert Fathi.

Expansiveness,
hyperbolicity
and Hausdorff dimension. Comm. Math. Phys.
126 (1989), no. 2, 249-262

*Finiteness of
Hausdorff dimension for expansive systems
and hence for topological dimensions and
other nice things.*

**Decay of correlations in dynamical systems**

Carlangelo Liverani, Central
limit
theorem for deterministic systems (1995)

Matt Nicol's probability lecture on Friday
afternoon derived the CLT using martingale
approximations: this paper gives more
details of this approach.

Dong Han Kim,

The
dynamical
Borel-Cantelli lemma for interval maps,
DCDS

**17** (2007), 891-900.

*The
Borel-Cantelli lemma is another result from
probability theory that can sometimes be
proved in a dynamical setting; given a
sequence of events, it addresses the
question of whether finitely many or
infinitely many of them occur.*

Carlangelo
Liverani, Decay
of
correlations in piecewise expanding maps,
Journal of Statistical Physics **78 **(1995),
p. 1111-1129.

This carries out the details of the proof of
decay of correlations using the method of
cones and the Hilbert metric, which will be
discussed in the lectures by Will Ott.

Liverani,
Carlangelo; Saussol, Benoit; Vaienti, Sandro.
A
probabilistic approach to intermittency.
Ergodic Theory Dynam. Systems 19 (1999), no.
3, 671-685.

*The
Manneville-Pomeau map was mentioned as an
example of a non-uniformly hyperbolic
dynamical system, which displays
"intermittent" chaotic behaviour; this is
studied in this paper.*

Multiplicative ergodic theory and applications
Sebastian Gouezel and Anders Karlsson:

Subadditive
and multiplicative ergodic theorems

Mark Pollicott: Maximal
Lyapunov exponents for random matrix
products. Invent. Math. 181 (2010)

*Fast estimates
of Lyapunov exponents for products of
matrices with positive entries, using
nuclear operators.*

Lai-Sang Young,

Ergodic
theory
of differentiable dynamical systems. Real
and complex dynamical systems (Hillerod, 1993),
293-336, NATO Adv. Sci. Inst. Ser. C Math. Phys.
Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995.

*A nice survey
paper.*

Jairo Bochi:

Notes
on a theorem of Furstenberg giving a criterion
for positive exponents.

**Poincare sections for diagonal maps**

A. Wright, From
Rational
Billiards to Moduli Spaces, Bull. Amer.
Math. Soc. (2016)

Survey article that gives a description of
the broader context for the lectures. If you
want further (much more detailed) reading,
consider: H. Masur and S.
Tabachnikov, Rational billiards and
Flat surfaces, Handbook of dynamical
systems, Vol. 1A, North-Holland, Amsterdam,
2002, pp. 1015-1089. Also A.
Zorich, Flat
surfaces, Frontiers in number theory,
physics and geometry. I, Springer, Berlin,
2006, pp. 437-583.

Y. Cheung,

Hausdorff
dimension
of the set of points on divergent trajectories
of a homogeneous flow on a product space,
Erg. Th. Dyn. Sys., 27 (2007), 65--85.

*Hausdorff
dimension itself is not discussed in the
lectures, but you can read this paper with
the goal of understanding the underlying
combinatorial models: continued fractions
and best approximations.*

J. Athreya and Y.
Cheung, A
Poincare section for the horocycle flow on
the space of lattices, Int. Math. Res.
Not., 2014 no. 10 (2014), 2643-2690.

J. S. Athreya, J.
Chaika, and S. Lelievre. The
gap distribution of slopes on the golden L.
In Recent trends in ergodic theory and
dynamical systems, volume 631 of Contemp.
Math., pages 47-62. Amer. Math. Soc.,
Providence, RI, 2015.