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Details of current and previous Dynamical Systems Seminars |
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Details of current and previous Dynamical Systems Seminars |
Ergodic Theory and Dynamical Systems Seminars
take place on Tuesdays between 2 and 3 PM,
in B3.03 (unless otherwise indicated) in the
Mathematics Institute.
For further information about Ergodic Theory and Dynamical Systems Seminars, contact the organisers Adam Epstein and Lasse Rempe.
Seminar talks during Term 1
Seminar talks during Term 2
Seminar talks during Term 3
5 October
Anthony Manning (Warwick)
A map of the tetrahedron that represents taking the pedal triangle
Abstract
The feet of the altitudes of a triangle are the vertices of a new triangle called its pedal triangle. But what happens to its shape if we take the pedal triangle of that, and repeat the process indefinitely?
Following work of Kingston-Synge, Lax, Ungar, and Hitt-Zhang we shall describe an expanding map of the tetrahedron whose orbits represent the angles of each triangle in such a pedal sequence. If the angles of the original triangle are an integer number of degrees then the sequence is periodic; but typically the pedal sequence approximates each possible shape and is only acute one quarter of the time!
We shall include some background from Euclidean geometry for those too young to have studied that!
12 October
Henk Bruin (Surrey)
Renormalization for interval translation maps
Abstract
In joint work with Serge Troubetzkoy (Luminy), we consider a family of interval translation map T on the circle, for which a renormalization operator (comparable to the Gauss map for rotations of the circle) exists. We show that for an uncountable set of parameters, T admits an attracting Cantor set, and show properties concerning the Hausdorff dimension and unique ergodicity of these Cantor sets.
19 October
Steve Kalikow
Does twofold mixing imply threefold mixing?
It is an open problem whether, for measure-preserving
,
26 October
Bas Lemmens
(Warwick)
Dynamics of sup-norm non-expansive maps
Abstract
The aim of this talk is to discuss the iterative behaviour of sup-norm nonexpansive maps. We shall describe various classes of sup-norm nonexpansive maps, indicate connections with monotone dynamical systems, and mention various results and open problems. In particular, we shall explain some ideas for a conjecture of Nussbaum concerning the periods of periodic points of sup-norm nonexpansive maps.
2 November
Mike Field (Houston)
Stability of rapid mixing for hyperbolic flows
Abstract
We describe recent work with Ian Melbourne (Surrey) and Andrew Torok (Houston) proving the genericity of C^2 stability of rapid mixing for Axion A flows. These results are improved to C^1 stability for Anosov flows and hyperbolic attractors.
Wednesday, 10 November, 2:00, room B1.01
David Sauzin (CNRS-Paris)
Resurgence of parabolic curves in C^2
(joint work with Vassili Gelfreich)
Abstract
A parabolic germ of analytic diffeomorphism of $(C^2,0)$ always admits invariant curves which may be called stable and unstable manifolds. We show, under some non-degeneracy hypothesis, how two such curves can be obtained as Borel sums of a single formal series in~$z^{-1}$ and~$z^{-1}\log z$, where~$z$ is a large variable.
Generically, this series is divergent and the two sums are not the analytic continuation one of the other; the leading order of their difference, which is exponentially small, is determined by a pair of complex constants. We analyse this phenomenon in the framework of Resurgence theory and prove that these constants depend analytically on parameters. They vanish for the time-1 map of an analytic vector field but not for a generic map.
(The seminar is exceptionally taking place on Wednesday, so as not to conflict with the David Fowler Memorial Symposium )
16 November
S. G. Dani (Tata Institute, Mumbai)
Asymptotic behaviour under application of random linear transformations
23 November
Phil Rippon (Open University)
Singularities of meromorphic functions with Baker domains
Abstract
We show that if f is a transcendental meromorphic function with a finite number of poles and f has a cycle of Baker domains of period p, then there exists C>1 such that the set of inverse function singularities of f^{-p} meets all sufficiently large annuli of the form {z: r/C < |z| < Cr}. This generalises a result of Bargmann on entire functions. The proof uses a technique developed by Eremenko and Lyubich for showing that the absence of such singularities forces the function to be 'expanding'. We also give examples to show that this result fails for transcendental meromorphic functions with infinitely many poles.
30 November
Ian Melbourne (Surrey)
A C^\infty Diffeomorphism with Infinitely Many
Intermingled Basins
Abstract
In joint work with Alistair Windsor (Austin, Texas), we construct a C^\infty diffeomorphism with infinitely many attractors A_1,A_2,... whose basins are intermingled in the following sense: Let B(A_j) consists of points whose omega-limit set is A_j. Then the union of the sets B(A_j) has full Lebesgue measure, but every open set has positive measure intersection with each B(A_j). The construction is given on the four dimensional compact manifold M = T^2 x S^2.
(There will be no seminar on Dec. 7 since the Workshop on Holomorphic Dynamics is taking place that week.)
18 January
Thomas Prellberg (Queen Mary)
Cluster Approximation for the Farey Fraction Spin Chain
Abstract
The Farey fraction spin chain is a number-theoretically motivated statistical mechanical model with long-range interactions and an interesting phase transition in the thermodynamic limit. We study the phase diagram of this spin chain coupled to an external field. With the help of a suitable transfer operator, we show that this spin chain is related to intermittent dynamics of an interval map. Utilizing a first-return map formalism, the thermodynamic behavior of the model is found in an effective cluster energy approximation. The results are consistent with renormalization group arguments and scaling theory.
25 January
David Chillingworth (Southampton)
Geometry of bifurcation from a manifold (with application to
coupled oscillators)
Abstract
The degeneracy inherent in bifurcating from a manifold can be addressed using ideas of genericity and stability from singularity theory. We illustrate some of these ideas in the context of single and coupled van der Pol oscillators.
1 February
Lasse Rempe (Warwick)
On a question of Eremenko concerning escaping sets of entire
functions
Abstract
(Joint work with G. Rottenfusser.) In 1926, Fatou observed that Julia sets of certain explicit transcendental entire functions contain (analytic) curves to infinity, consisting of points tending to infinity under iteration, and states that it would be interesting to know whether this holds in much greater generality. In 1984, Eremenko asked whether every escaping point of a transcendental entire function can be connected to infinity by such a curve. These questions are of particular importance since a positive answer would allow the dynamical study of these functions by combinatorial means, much as in the theory of "external rays" for polynomials. We give a positive anser to Eremenko's question for a large class of functions, including all finite-order entire function with a bounded set of singular values, and any finite composition of such functions.
8 February
Eleanor Clifford (Loughborough)
Recent results in normal families
Abstract
We consider some recent results in normal families, and some of the methods used in these proofs. In particular, we will look at the Bloch Principle, the Pang-Zalcman Lemma and some theorems concerning value distribution.
15 February
Mariannne Freiberger (Queen Mary)
A family of matings between entire transcendental functions and a Fuchsian group
Abstract
We describe the dynamics of a family of holomorphic correspondences which represent matings between scalar multiples of the function z-> sin z and a Fuchsian group.
22 February
Luca Sbano (Warwick)
Choreographic motions in molecular potentials
Abstract
We shall consider a system of three equal particles in a plane, interacting through a potential of Lennard-Jones type and we show, by means of critical point theory, that such systems admit "choreographic" solutions: three particles chasing each other along a single curve. For each homotopy class of choreographies we prove the existence of at least one solution. This solution is a mountain-pass critical point of the action functional.
1 March
Samuel Lelievre (Ecole Polytechnique)
Square-tiled surfaces in genus two
Abstract
On the moduli spaces of abelian differentials exists a natural action of SL(2,R). Its orbits, called Teichm\"uller discs, project in the moduli spaces of Riemann surfaces to complex geodesics. Pulling back the form dz of the standard torus by coverings branched over a single point, one obtains the square-tiled surfaces, integer points of the moduli spaces of abelian differentials. We study in detail the Teichm\"uller discs of integer points of the moduli space of abelian differentials in genus two with a double zero: number of Teichm\"uller discs for each number of square tiles, and their geometry; algebraic properties of the stabilisers (subgroups of SL(2,Z) which are not congruence subgroups); asymptotic behaviour of the Siegel-Veech constants (coefficients of the quadratic growth rates of closed geodesics) when the number of tiles tends to infinity.
8 March
Edward Crane (Oxford)
Invariant and balanced measures for holomorphic correspondences
Abstract
Let $C$ be a smooth curve over the complex numbers. An irreducible holomorphic correspondence is the relation induced by a (possibly singular) curve $D$ in $C\times C$. We study the dynamics that arise from iterating a holomorphic correspondence. Even in trying to define an analogue of the Julia set, various sensible definitions are possible, leading to different limit sets. Rather than concentrate on topological dynamics, we can begin by studying the measurable dynamics. With reference to subshifts of finite type and to the measurable dynamics of rational maps, I will discuss what is the correct notion of an invariant measure for a holomorphic correspondence. I will describe some recent results about invariant measures and illustrate with examples.
26 April
NO SEMINAR
3 May
Mark Holland (Surrey)
Stable manifolds and hyperbolic systems
Abstract
As an introduction, I will disucss the role stable manifolds play in understanding local and global dynamics. In particular I will discuss their role in ergodic theory, especially in the analysis of "basin" problems for strange attractors (like the Henon and Lorenz attractors). In the non-uniformly hyperbolic setting, I will describe useful techniques in constructing stable manifolds.
10 May
NO SEMINAR
17 May
Oscar Bandtlow (Nottingham)
Bounded distortion versus decaying derivatives
Abstract
Let T be a real analytic full branch expanding map. A well known sufficient condition for the existence of a T-invariant probability measure equivalent to Lebesgue measure is that T has bounded distortion. In this talk I will discuss an independent sufficient alternative: the decaying derivatives condition.
24 May
Thomas Ward (East Anglia)
Entropy and topological rigidity
This talk describes the connection between topological entropy and topological rigidity for mixing actions on connected groups. This rigidity property forces continuous maps that intertwine two algebraic dynamical systems to be affine maps. This is joint work with Siddhartha Bhattacharya.
31 May
Konstantin Khanin (Herriot-Watt)
Renormalizations, multi-dimensional continued fractions and KAM theory.
7 June
Michael Todd (Surrey)
TBA
21 June
Simon Lloyd (Warwick)
The C^r Closing Problem for vector fields on the torus
Abstract
Given a recurrent point p of a vector field X, is there a vector field Y close to X (in some topology) for which the orbit of p is periodic? On compact manifolds, Pugh's C^1 Closing Lemma shows the answer is 'yes' for C^1 vector fields with the C^1 topology. For r>1, the corresponding C^r Closing problem is still open. In this talk I shall explain some partial results in this direction for vector fields on the torus.