2017 Houston Summer School on Dynamical Systems2017 UH Summer school in dynamics: Project options Statistical
properties in hyperbolic dynamics 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. S2. C. Liverani, Central limit theorem for deterministic systems (1995) 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 was discussed in Will Ott's lectures. The second extends this to some non-uniformly hyperbolic dynamical systems, which display "intermittent" chaotic behavior. Partially hyperbolic dynamics P1. An expository paper on results about stable ergodicity. Burns, Keith ; Pugh, Charles ; Shub, Michael ; Wilkinson, Amie. Recent results about stable ergodicity. Smooth ergodic theory and its applications (Seattle, WA, 1999), 327-366, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001. P2. Bonatti, Christian ; Matheus, Carlos ; Viana, Marcelo ; Wilkinson, Amie. Abundance of stable ergodicity. Comment. Math. Helv. 79 (2004), no. 4, 753-757. Abstract: We
consider the set of volume preserving
partially hyperbolic diffeomorphisms on a
compact manifold having \(1\)-dimensional
center bundle. We show that the volume
measure is ergodic, and even Bernoulli, for
any \(C^2\) diffeomorphism in an open and
dense subset. This solves a conjecture of
Pugh and Shub, in this setting. P3. Burns, Keith ; Hertz, Federico Rodriguez ; Hertz, Maria Alejandra Rodriguez ; Talitskaya, Anna ; Ures, Raul. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 75-88. Abstract: It
is shown that stable accessibility property
is \(C^r\)-dense among partially hyperbolic
diffeomorphisms with one-dimensional center
bundle, for \(r\ge 2\), volume preserving or
not. This answers a conjecture by Pugh and
Shub for these systems. P4. Wilkinson, Amie. Conservative partially hyperbolic dynamics. Proceedings of the International Congress of Mathematicians. Volume III, 1816-1836, Hindustan Book Agency, New Delhi, 2010. Abstract: We
discuss recent progress in understanding the
dynamical properties of partially hyperbolic
diffeomorphisms that preserve volume. The
main topics addressed are density of stable
ergodicity and stable accessibility, center
Lyapunov exponents, pathological foliations,
rigidity, and the surprising
interrelationships between these notions. Dynamical methods in Diophantine approximation D1. Bourgain, Jean ; Lindenstrauss, Elon ; Michel, Philippe ; Venkatesh, Akshay. Some effective results for \(\times a\times b\). Ergodic Theory Dynam. Systems 29 (2009), no. 6, 1705-1722. The authors
prove quantitative results on the rate at
which the sequence \(\{a^nb^kx\}\) becomes
dense, for \(a, b\) fixed multiplicatively
independent integers and
\(x\in\mathbb{R}/\mathbb{Z}\). The proofs
use techniques from entropy theory and
harmonic analysis, together with lower
bounds for linear forms in logarithms due to
Baker and Wustholz. D2. Einsiedler, Manfred ; Fishman, Lior ; Shapira, Uri. Diophantine approximations on fractals. Geom. Funct. Anal. 21 (2011), no. 1, 14-35. Focus on
Theorems 1.5 and 1.11, which use dynamics in
the space of unimodular lattices to derive
what seem to be highly non-trivial results
about Diophantine approximation properties
of integer multiples of real numbers. D3. Einsiedler, Manfred ; Ward, Thomas. Ergodic theory with a view towards number theory, Chapter 9: Geodesic flow on quotients of the hyperbolic plane. Graduate Texts in Mathematics, 259. Springer-Verlag London, Ltd., London, 2011. This chapter
provides a very clear introduction to the
basics of dynamics and ergodic theory in
closed linear groups and their quotients.
Some of the highlights are a proof of the
ergodicity of the geodesic flow on quotients
of \(\mathrm{PSL}_2(\mathbb{R})\) by
lattices, and a detailed explanation of the
connection between the geodesic flow and the
Gauss map on the unit interval. D4. Host, Bernard. Nombres normaux, entropie, translations. Israel J. Math. 91 (1995), no. 1-3, 419-428. Abstract:
Given a measure \(\mu\) on the circle, we
study the relations between the entropy of
the multiplication by an integer p and the
conservativity for the translations by the
\(p\)-adic rational numbers. We get a
criterion for \(\mu\)-almost every point to
be normal in a basis \(q\) prime to \(p\),
and generalizations of the result of D.
Rudolph about measures which are invariant
by multiplication by \(p\) and \(q\). Dynamics of group actions on homogeneous spaces H1. Bombieri, E. ; Vaaler, J. On Siegel's lemma. Invent. Math. 73 (1983), no. 1, 11-32. The authors
develop an adelic formulation of Minkowski's
theorems with nice applications. For
background on the ring of adeles, see Andre
Weil's book "Basic Number Theory". H2. Howe, Roger ; Tan, Eng-Chye. Nonabelian harmonic analysis, Applications of SL(2,R), Chapter 5: Asymptotics of matrix coefficients. Universitext. Springer-Verlag, New York, 1992. This
involves a study of basic properties of
unitary representations of \(\mathrm{SL}(2,
\mathbb{R})\), the Harish-Chandra function
of \(\mathrm{SL}(2, \mathbb{R})\)
culminating in a proof of quantitative
estimates for decay of matrix coefficients. H3. Maucourant, Francois. Dynamical Borel-Cantelli lemma for hyperbolic spaces. Israel J. Math. 152 (2006), 143-155. Abstract:
This paper studies shrinking target
properties for geodesic flows with targets
inside a hyperbolic manifold. H4. Sullivan, Dennis. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149 (1982), no. 3-4, 215-237. This paper
develops logarithm laws using mixing, but in
a different, more geometric way than the
approach used in the lectures. Linearly recurrent systems L1. Three papers having to do with dynamics on tiling spaces. Robinson, E. Arthur, Jr. Symbolic dynamics and tilings of \(\mathbb{R}^d\). Symbolic dynamics and its applications, 81-119, Proc. Sympos. Appl. Math., 60, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 2004. An
introduction to the study of Euclidean
tilings through the use of dynamical systems
and ergodic theory. B. Solomyak. Dynamics of self-similar tilings. Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695-738. A
generalization to the tiling setting of
substitutive symbolic dynamics. An
introduction to dynamical systems arising
from the action by translations on the orbit
closures of self-similar and self-affine
tilings of \(\mathbb{R}^d\). The main focus
is on spectral properties of such systems
which are shown to be uniquely ergodic. M. I. Cortez, F. Durand, B. Host, A. Maass. Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems. J. London Math. Soc. (2) 67 (2003), no. 3, 790-804. This paper
investigates the connection between
topological and measure-theoretical
eigenvalues for linearly recurrent Cantor
systems. L2. Three papers having to do with linearly repetitive Delone sets. J. C. Lagarias, Peter A. B. Pleasants. Repetitive Delone sets and quasicrystals. Ergodic Theory Dynam. Systems 23 (2003), no. 3, 831-867. An extension
to the setting of Delone sets of the study
of factor complexity and recurrence. J. Aliste-Prieto, D. Coronel, M. I. Cortez, F. Durand, S. Petite. Linearly repetitive Delone sets. Mathematics of aperiodic order, 195-222, Progr. Math., 309, Birkhauser/Springer, Basel, 2015. An
introduction to linearly repetitive Delone
sets and their associated dynamical systems. A. Haynes, H. Koivusalo, J. Walton. Linear repetitivity and subadditive ergodic theorems for cut and project sets. arXiv:1503.04091. An extension
of results on the characterization of linear
recurrence for Sturmian words to the
cut-and-project setting. L3. F. Durand. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1061-1078, corrigendum and addendum in ETDS 23 (2003), no. 2, 663-669. M. I. Cortez, F. Durand, S. Petite. Linearly repetitive Delone systems have a finite number of nonperiodic Delone system factors. Proc. Amer. Math. Soc. 138 (2010), no. 3, 1033-1046. The first
paper proves the characterization of
linearly recurrent subshifts as primitive
and proper \(S\)-adic subshifts. The
second extends these symbolic results to
linearly repetitive Delone sets.
Funding for this event is
provided by NSF grants DMS-1554794 and
DMS-1700273.
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