One dimensional quantum Ising model and hyperbolic dynamics

Speaker: 

William Yessen

Institution: 

UC Irvine

Time: 

Tuesday, October 12, 2010 - 3:00pm

Location: 

RH 440R

The one dimensional quantum Ising model is used in quantum statistical physics to model interracting particles on a discrete lattice. While the classical model (in one and two dimensions) has long been solved (its origin dates back to 1930's), its quasiperiodic analog (dating back about 25 years) is still a source of interesting problems. We shall discuss our solution to one such problem: we'll rigorously prove that the energy spectrum of the one dimensional quantum quasiperiodic Ising model is a Cantor set, as has been long believed, and discuss some of its properties.
This is the first in a series of two seminars dedicated to this topic. In this seminar we'll present the problem and set up the main ideas.

On conservative homoclinic bifurcations: recent results and open problems

Speaker: 

Anton Gorodetski

Institution: 

UC Irvine

Time: 

Tuesday, October 5, 2010 - 3:00pm

Location: 

RH 440R

In 1970s S.Newhouse discovered that a generic homoclinic bifurcation of a smooth surface diffeomorphism leads to persistent homoclinic tangencies, infinite number of attractors (or repellers), and other unexpected dynamical properties (nowadays called "Newhouse phenomena"). More than 20 years later P.Duarte provided an analog of these results in conservative setting (with attractors replaced by elliptic periodic points). We will discuss these and other recent results on conservative homoclinic bifurcations, and list some related open problems in the field.

On one-dimensional Schrodinger Operator with potential given by Period Doubling and Thue-Morse sequences.

Speaker: 

Vinod Sastry

Institution: 

UC Irvine

Time: 

Friday, June 4, 2010 - 2:00pm

Location: 

RH 440R

We consider the one-dimensional Schrodinger operator with potential given by Period Doubling and Thue-Morse sequences. We describe the results of Bellisard et al. in which the spectrum of these operators are obtained through the trace map associated to this operator. We see that the spectrum for non-zero values of the potential is a Cantor set of zero Lebesque measure, and give explicit description of the spectral gaps.

Application of trace map techniques to 1D Ising models, II

Speaker: 

William Yessen

Institution: 

UC Irvine

Time: 

Friday, April 23, 2010 - 2:00pm

Location: 

RH 440R

We consider the classical 1D Ising model, where the coupling constants and the external magnetic field take one of two possible values at each site, according to a substitution rule. We shall introduce (briefly) the idea of a trace map corresponding to the given substitution rule and how its dynamical properties can be used to investigate the partition function and, consequently, the free energy function of the given Ising model.

Application of trace map techniques to 1D Ising models

Speaker: 

William Yessen

Institution: 

UC Irvine

Time: 

Friday, April 16, 2010 - 2:00pm

Location: 

RH 440R

We consider the classical 1D Ising model, where the coupling constants and the external magnetic field take one of two possible values at each site, according to a substitution rule. We shall introduce (briefly) the idea of a trace map corresponding to the given substitution rule and how its dynamical properties can be used to investigate the partition function and, consequently, the free energy function of the given Ising model.

Hyperbolic geometry of ultrametric spaces

Speaker: 

Zair Ibragimov

Institution: 

CalState Fullerton

Time: 

Friday, April 30, 2010 - 2:00pm

Location: 

RH 440R

We will discuss basic properties of ultrametric spaces. Well-known examples of complete ultrametric spaces are p-adic numbers as well as p-adic integers. Also, it is known that the boundary at infinity of metric trees as well as more general Gromov 0-hyperbolic spaces is a complete bounded ultrametric space when equipped with a visual metric. We will discuss this result in details and show that the converse statement also holds. Namely, we show that every complete ultrametric space arises as the boundary at infinity of both a Gromov 0-hyperbolic space as well as a metric tree.

Measures of maximal entropy for some robustly transitive diffeomorphisms

Speaker: 

Todd Fisher

Institution: 

Brigham Young University

Time: 

Friday, May 21, 2010 - 2:00pm

Location: 

RH 440R

Dynamical entropies are measures of the complexity of orbit structures. The topological entropy considers all the orbits, whereas the measure theoretic entropy focuses on those ``relevant" to a given invariant probability measure. The variational principle says that the topological entropy of a continuous self-map of a compact metrizable space is the supremum of the measure theoretic entropy over the set of invariant probability measures for the map.

A well known fact is that every transitive hyperbolic (Anosov) diffeomorphism has a unique invariant probability measure whose entropy equals the topological entropy. We analyze a class of deformations of Anosov diffeomorphisms containing many of the known nonhyperbolic robustly transitive diffeomorphisms. We show that these $C0$-small, but $C1$-macroscopic, deformations leave all the high entropy dynamics of the Anosov system unchanged, and that there is a partial conjugacy identifying all invariant probability measures with entropy close to the maximum for the deformation with those of the original Anosov system.

Additionally, we show that these results apply to a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana and a class originally described by Mane. In fact these methods apply to several classes of systems which are similarly derived from Anosov, i.e., produced by an isotopy from an Anosov system.

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