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Brownian Subordinators And Fractional Cauchy Problems
by
Boris Baeumer
University of Otago
Coauthors: M.M. Meerschaert and E. Nane (both at Michigan State University)
A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. We show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations.
Date received: October 26, 2007
Ramsey theory from a dynamical viewpoint
by
Mathias Beiglböck
TU Vienna
Coauthors: Vitaly Bergelson, Neil Hindman, Dona Strauss
Theorems in Ramsey theory are formalizations of the principle that highly organized structures are unbreakable. For instance, van der Waerden's Theorem states that one cell of any finite partition of the integers contains arithmetic progressions of arbitrary finite length. Following seminal papers of Furstenberg, methods from ergodic theory and topological dynamics have been applied to give short proofs to such classical results as well as to solve various open problems. We discuss this abstract approach and present an extension of van der Waerden's Theorem which refers simultaneously to the additive and the multiplicative structure of the integers. (Supported by the Austrian Science Foundation FWF, project no. S9612)
Date received: October 30, 2007
Uniform attraction and growth in nonautonomous dynamical systems
by
Arno Berger
University of Alberta, Canada and University of Canterbury, NZ
Uniformity plays an important role in nonautonomous dynamics; for parts of this notoriously heterogeneous discipline, most prominently perhaps for the emerging theory of nonautonomous bifurcations, it is in fact quite indispensable. This talk will discuss some of the more subtle, less expected implications of uniformity (in time) pertaining to two topics of considerable current interest in nonautonomous dynamics: the natural concept of uniform attractors/repellors, and the foundations of dynamic partitions or, more generally, finite time dynamics. (Joint work with T.S. Doan and S. Siegmund.)
Date received: October 30, 2007
Typical partially hyperbolic diffeomorphisms with one dimensional center are accessible
by
Keith Burns
Northwestern University
Coauthors: Jana and Federico Rodriguez Hertz, Anna Talitskaya, Raul Ures
I will outline how this result fits into Pugh and Shub's program for studying the ergodic theory of partially hyperbolic diffeomorphisms and sketch the main ideas in the proof.
Date received: October 31, 2007
Galton board
by
Dmitry Dolgopyat
University of Maryland, USA
Coauthors: Nikolai Chernov
Galton board, is a device invented by Sir Francis Galton to demonstrate the law of error and the normal distribution. The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce randomly left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.
We assume that the collisions of the ball with the pins are perfectly elastic and that there is no friction. We show that the ball almost surely exits the machine from the hole at the top before reaching the bins at the bottom provided that we have sufficiently many rows of pins.
Date received: October 25, 2007
The AT property is not preserved by finite extensions
by
Anthony Dooley
UNSW, Sydney
Coauthors: Anthony Quas
The property of almost transitivity was introduced by Connes and Woods to characterise flows arising from actions of infinite product type.
A question was posed by Giordano, Putnam and Skau as to whether this property was preserved under finite extensions. A simple example was whether the Morse system was AT, as it can be realised as a two-point extension of an infinite product system. In this work, we show that the Morse system is AT, but produce a two-point extension of an AT system which is not AT. This has consequences for operator algebras: there exists an ITPF1 factor with an index two subfactor which is not ITPF1.
Date received: October 30, 2007
Phase transitions and equilibrium states
by
Gary Froyland
School of Mathematics and Statistics, University of New South Wales
Coauthors: Dalia Terhesiu (UNSW) and Rua Murray (Canterbury)
It is well known that for several classes of transformations, Ulam's method is an efficient way to estimate the absolutely continuous invariant measure of T. We describe a new extension of Ulam's method that can be used for the numerical approximation of the Ruelle-Perron-Frobenius operator associated with T and the standard potential fb=- b log|T\shortmid|, where b ∈ R. In particular we demonstrate that our extended Ulam's method is a powerful tool for computing the topological pressure P(T, fb) and the density of the equilibrium state. We state convergence results, illustrate our approach via examples and demonstrate its effectiveness, even when applied to nonuniformly expanding maps. This work complements recent analytical studies of the statistical properties of nonuniformly expanding maps by offering a simple, fast, and accurate numerical tool for the analysis of Ruelle-Perron-Frobenius operators and their associated thermodynamical objects.
Date received: November 11, 2007
Equidistribution of closed geodesics on the modular surface
by
Wenzhi Luo
The Ohio State University
Coauthors: Zeev Rudnick and Peter Sarnak
It is well-known that the closed geodesics on the modular surface, when collected according to the discriminants, are equidistributed with respect to the hyperbolic measure, by the works of Duke and Iwaniec. We evaluate asymptotically the variance of this distribution on the unit tangent bundle, and show it is equal to the classic variance of the geodesic flow as studied by Ratner, multiplied by an intriguing arithmetic invariant, the central value of certain L-function. Our approach is via Weil representation and the theta correspondence. This is the joint work with P.Sarnak and Z.Rudnick.
Date received: October 29, 2007
Decay of correlations for Lorentz gases
by
Ian Melbourne
University of Surrey
In this talk, I will describe some recent results on decay of correlations for various Lorentz gas models, including infinite horizon Lorentz gases, Bunimovich stadia, and cuspoidal domains.
The cuspoidal example (joint work with Balint, hopefully finished in time) is particularly interesting because we are proving superpolynomial decay for the flow even though the collision map (billiard map) mixes very slowly.
Date received: October 28, 2007
Ulam's method for invariant measures with an indifferent fixed point
by
Rua Murray
University of Canterbury
Ulam's method is now a well-known technique for gaining numerical access to invariant densities for uniformly expanding maps. However, convergence analyses for the approximations have usually relied on a strong spectral picture for the Frobenius-Perron operator (for example, quasi-compactness in BV for uniformly expanding maps). Even in the case of an interval map which is strictly expanding except at a single point, more delicate analysis is needed. Ideas from Young's tower constructions can be adapted to show that in the case of an indifferent fixed point with tangency of order x1+a (0 < a < 1), the Ulam approximately invariant densities converge in L1 as finer grids are used. An explicit convergence rate (depending on a) will be given.
Date received: October 30, 2007
Extreme value statistics for non-uniformly hyperbolic systems
by
Matthew Nicol
Mathematics Department, University of Houston
Coauthors: Mark Holland (University of Exeter)
Andrew Torok (University of Houston)
Suppose ft: X → X is a non-uniformly hyperbolic map (discrete-time) or flow (continuous time) which may be modelled by a Young tower. Suppose f: X → R is a function on X which is locally Holder except for a finite number of singular points. Define Zt(x)=max0 ≤ s ≤ t{ fs(x)}. We show that the possible nondegenerate limit distributions for Zt under linear scaling are the type I, II and III distributions of extreme value statistics. We also determine which particular distribution arises (I, II or III) as a function of the regularity of f and the underlying dynamics.
Date received: October 29, 2007
Distances in positive density sets
by
Anthony Quas
University of Victoria
Given a set of distances D, one can consider the graph Gd, D on Rd where two points are adjacent if they are separated by a distance belonging to D and ask for its chromatic number. The case where D={1} is the Hadwiger-Nelson problem and it is known that 4 ≤ c(G2, {1}) ≤ 7. If the colour classes are required to be measurable, we obtain the measurable chromatic number cm(Gd, D). It is known that 5 ≤ cm(G2, {1}) ≤ 7.
In the case where D is unbounded, it turns out that cm(Gd, D)=∞. We give a conceptual new proof of this and discuss possible extensions to the general (non-measurable) case.
Date received: October 24, 2007
Spectra of Ruelle transfer operators for contact flows on basic sets
by
Luchezar Stoyanov
University of Western Australia
This talk concerns contact flows on Riemann manifolds satisfying a certain pinching condition over a basic set. Under some additional geometric conditions on the basic set (always satisfied e.g. when the flow is Anosov or when the stable and unstable laminations are one- dimensional), strong spectral estimates are obtained for the Ruelle transfer operators related to arbitrary (Hölder continuous) potentials. These estimates are similar to the ones proved by Dolgopyat in the case of Anosov flows with smooth jointly non-integrable stable and unstable foliations. As is well-known, such estimates lead to some intersting consequences such as the existance of a non-trivial meromorphic extension of the (Ruelle) dynamical zeta function and exponential decay of correlations for the flow over the given basic set.
Date received: October 28, 2007
Canard induced mixed-mode oscillations
by
Martin Wechselberger
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Coauthors: Nancy Kopell (University of Boston),
Horacio Rotstein (NJIT),
Warren Weckesser (University of Sydney)
Mixed-mode oscillatory temporal patterns (MMOs) consist of a combination of subthreshold oscillations and spikes. We present a model of medial entorhinal cortex stellate cells (SC model) and show that the mechanism responsible for the observed subthreshold oscillations is based on the canard phenomenon. We explain the canard theory in detail and show which ionic currents are responsible for this phenomenon in the SC model. In particular, we show how variations of key parameters cause bifurcations of MMO patterns.
Date received: October 25, 2007
Tilings and Gallai's Theorem
by
Alistair Windsor
University of Memphis
Coauthors: Rafael de la Llave (University of Texas at Austin)
We will discuss tilings of the plane, concentrating on aperiodic tilings of the plane of finite local complexity, such as the Kite and Dart tiling of Penrose, or the remarkable Pinwheel tiling due to Conway and Radin. The Penrose tiling can be seen in Storey Hall at the Royal Melbourne Institute of Technology. The Pinwheel tiling can be seen in Melbourne's Federation Square. Using combinatorics, or its equivalent statement in topological dynamics, we prove a result about the appearance of certain configurations.
Date received: September 21, 2007
For a topologist, typical sequences are extremely irregular
by
Reinhard Winkler
TU Wien (University of Technology Vienna, Austria)
Coauthors: Martin Goldstern, Joerg Schmeling
From the measure theoretic point of view the typical distribution behavior of sequences is regular in the sense of the law of large numbers (implying uniform distribution) and other main results from probability and ergodic theory. The world looks totally different from the topological point of view where, instead of sets of measure 0, meager sets are considered to be negligible. For instance, most sequences in a compact metric space (i.e. all sequences with the exception of a meager subset) are what we call maldistributed, the extreme opposite of being distributed according to one measure. I present several statements of this flavour including a topological counterpart of Birkhoff's ergodic theorem for transitive dynamical systems. (Supported by the Austrian Science Foundation FWF, project no. S9612)
Date received: October 29, 2007
Nonmonotonicity of phase transitions in a tree loss network
by
Ilze Ziedins
The University of Auckland
Coauthors: Brad Luen (Berkeley), Kavita Ramanan (Carnegie Mellon)
We consider a symmetric tree loss network that supports single-link and multi-link connections to nearest neighbours, with finite capacity C on each connecting link. Connections arrive as Poisson processes and have generally distributed holding times with finite mean. At sufficiently high multi-link arrival rates the network exhibits a phase transition, with multiple Gibbs measures existing on the infinite tree. When a simple control is introduced into the network, the phase transition is nonmonotone in the arrival rate of the multi-link connections.
Date received: October 30, 2007