2016 Houston Summer School on Dynamical Systems

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.