Workshop on Dynamical Methods in Spectral Theory of Quasicrystals

Department of Mathematics
University of California, Irvine

Thursday May 16 -- Sunday May 19, 2013

Where the workshop will take place
Travel, parking, lodging and dining information
Organizers and contacts
Confirmed participants
Abstracts of mini-courses
Abstracts of talks


University of California, Irvine, Room 1201 Natural Sciences 2.  See the map of UCI. Natural Sciences 2 is Building 402 on the map (located to the south of the Aldrich park, just beyond the outer circle surrounding the park).

Internet on campus:

Wireless internet access is provided everywhere on the UC Irvine campus for free. The network name is "UCI Mobile Access." You will need to register your computer's MAC address. These are the steps:

1) Connect to UCIMobile network.
2) Direct your browser to any webpage (e.g.
3) The system will redirect to express registration. This process is automatic and should not take more than three minutes (there will be a timer; while the timer is running, do not try to open any webpages or reload the current page, as this will just reset the timer).

In case of technical difficulties, contact Will Yessen (see the organizers section below).

Travel, Parking, Lodging and Dining:

Most participants will be accommodated in Radisson Newport Beach hotel. A scheduled shuttle to/from UCI will be provided.

For those flying to UC Irvine: the closest airport is the John Wayne airport in Santa Ana, California (airport code: SNA), located 3 miles from UC Irvine; Los Angeles International Airport (airport code: LAX) is located 43 miles from UC Irvine.

There are several choices for dining around UC Irvine as well as Radisson Newport Beach hotel.

Dining around UCI: see this map. In this map, we will be located by the UCI Science Library, which is to the west of the green circle (the Aldrich park) enclosed by the Campus Dr. and Peltason Dr.

Dining around Radisson Newport Beach hotel: see this map.

To see more choices on the maps above, on the left hand side, scroll down and click "next". A packet with printouts of the above maps will be distributed to participants at the meeting.


• David Damanik, Rice University, Department of Mathematics (damanik [at] rice [dot] edu);
• Anton Gorodetski, UC Irvine, Department of Mathematics (asgor [at] math [dot] uci [dot] edu);
• May Mei (mmei [at] math [dot] uci [dot] edu);
• Will Yessen (wyessen [at] math [dot] uci [dot] edu).

Confirmed Participants:

Name Affiliation
Juan Adame California Institute of Technology
Jean Bellissard Georgia Institute of Technology
David Damanik Rice University
Jake Fillman Rice University
Rui Han UC Irvine
Svetlana Jitomirskaya UC Irvine
Alexander Gordon University of NC at Charlotte
Anton Gorodetski UC Irvine
Silvius Klein Universidade de Lisboa, Portugal
Rostislav Kozhan UC Los Angeles
Michel L. Lapidus UC Riverside
Marius Lemm California Institute of Technology
Milivoje Lukic Rice University
Chris Marx California Institute of Technology
Rajinder Mavi University of Virginia
May Mei UC Irvine
Gabriel Maybrun UC Irvine
Paul Munger Rice University
Son Nguyen UC Irvine
John Rock CalPoly Pomona
Christophe Sabot Université Lyon 1
Christian Sadel University of British Columbia, Canada
Yuki Takahashi UC Irvine
Sidney Tsang UC Irvine
Will Yessen UC Irvine
Zhenghe Zhang Northwestern University
Maxim Zinchenko University of New Mexico


The schedule can be downloaded in PDF format here.

Abstracts of Mini-courses:

Wannier Transform and Bloch Theory for Aperiodic repetitive FLC tilings.
Jean V. Bellissard (Georgia Institute of Technology, School of Mathematics and School of Physics)
This mini-course will be divided into four sections:

1) Background: Delone sets, repetitivity, aperiodicity, finite local complexity, Hull, Transversal, the Lagarias group. Groupoids: the groupoid of the transversal, representation, covariance, covariant fields of Hilbert spaces, unitary equivalence.

2) The Anderson Putnam complex, examples (Fibonnaci, Thue-Morse, Rudin-Shapiro, the octagonal lattice, the Penrose lattice), inflations (for the substitutive case or the general case), inverse limits and reconstruction of the Hull. Substitution tilings. Bratteli diagrams, horizontal edges, reconstruction of the groupoid of the transversal.

3) The Wannier transform for one AP-complex: quasi-momentum space, the periodic case (as a reminder), the case of FLC tilings, the Plancherel formula (the Annier transform as a unitary transformation between covariant fileds of Hilbert spaces).

4) The momentum space decomposition of the Schrodinger operator. Description of the boundary conditions, cohomological equation. Examples in one dimension, the higher dimensional case. The finite volume approximation: band spectrum, absolute continuity. Discussion and conjectures: the infinite volume limit. Renormalization and inverse limit, the substitution case, expected estimates. A conjecture: absolutely continuous spectrum in D > 2 in the perturbative regime, a route towards a proof.
Trace Map Dynamics and spectral properties of the Fibonacci Hamiltonian.
David Damanik (Rice University) and Anton Gorodetski (UC Irvine)
1. Discrete one dimensional Schrodinger operators. Quantum dynamics, spectral measures, solutions and transfer matrices. Fibonacci Hamiltonian, recursions, Fibonacci Trace Map. Suto's Theorem.

2. Hyperbolic Dynamical Systems. Hyperbolicity of the Trace Map. Dynamically defined Cantor sets, their properties. Spectrum of the Fibonacci Hamiltonian as a dynamically defined Cantor set.

3. Fractal dimension of the spectrum. Gap labeling and gap opening. Transport exponents. Density of states measure. Square and cubic Fibanacci Hamitonian. Structure of spectrum and density of states measure.

4. Sums of Cantor sets and convolutions of singular measures. Absolute continuity of the density of states measure of square Fibonacci Hamiltonian in small coupling regime. Known results and open questions.
Spectral Theory of self-similar lattices and dynamics of rational maps.
Christophe Sabot (Université Lyon 1)
These lectures will focus on the spectral theory of "finitely ramified" self-similar lattices and their continuous counterparts, self-similar fractals (e.g. Sierpinski gasket, Snowflake, etc). These lattices, which can be viewed as toy-models for quasi-crystals, exhibit interesting relations with the dynamics of rational maps with several variables. The first lectures will be devoted to elementary properties of these lattices and to a short overview of multidimensional rational dynamics and pluripotential theory. Then, we will describe the renormalization map associated with a self-similar lattice and the crucial point of its appropriate compactification. We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neuman-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map the spectral properties of the operators are drastically different.
Spectral theory and dynamics of quasi-periodic Jacobi cocycles.
Svetlana Jitomirskaya (UC Irvine) and Chris Marx (California Institute of Technology)
Recently, several problems from the spectral theory of 1 dimensional quasi-periodic Jacobi operators could be successfully solved studying the dynamics of associated cocycles. Even though some of these ideas are well known in dynamical systems, they are still less common in the spectral theory community. In our lectures, we aim to bridge the gap, presenting a survey of the dynamics of quasi-periodic cocycles from a spectral theorists point of view.

Topics that will be discussed include: quasi-periodic Jacobi and higher-dimensional cocycles, Lyapunov exponent (LE) and Oseledets' theorem, Conjugacies - Aubry duality revisited, dominated splittings, (almost) reducibility and the absolutely continuous spectrum, complexified cocycles, continuity of the LE, Avila's global theory and spectral consequences. Our recent work on Extended Harper's Model, a generalization of almost Mathieu, will serve as an illustration for some of the presented methods.

Abstracts of Talks:

Abstracts of talks can be downloaded here.