# seminars0708

**2007-2008**

**Ergodic Theory and Dynamical Systems Seminars
** take place on

**Tuesdays between 2 and 3 PM, in B3.02**(unless otherwise indicated) in the

**Mathematics Institute**.

For further information about Ergodic Theory and Dynamical Systems Seminars, contact the organiser Carlos Cabrera. For a list of last years seminars see here.

9th October

** Vassili Gelfreich. (Warwick)**

**Diffusion-like dynamics in deterministic Hamiltonian systems **

16th October

** Freddie Exall (Liverpool) **

**An introduction to equivalent matings **

23th October

** Lasse Rempe (Liverpool) **

**Mane's theorem and absence of line-fields for transcendental meromorphic functions **

30th October

** Richard Sharp (Manchester) **

**Lengths, quasi-morphisms and limit laws **

6th November

** Yuki Yayama (East Anglia) **

**Dimensions of compact invariant sets of some expanding maps **

**Abstract:** We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski carpet" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by considering a general Sierpinski carpet represented by a shift of finite type. Applying results of Ledrappier-Young and Shin, we study the Hausdorff dimension of such a general Sierpinski carpet for the case when there is a saturated compensation function. We give some conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

13th November

** Carlos Cabrera (Warwick)**

**The inverse limit of z^2-1 **

20th November

** Alexander Bufetov (Rice University)**

**Existence and uniqueness of the measure of maximal entropy for the Teichmueller flow on the moduli space of abelian differentials (joint with B.M.Gurevich). **

**Abstract:** The moduli space of abelian differentials admits a natural Lebesgue measure class and a natural finite measure in that class, invariant under the Teichmueller flow. The talk will show this measure to be the unique measure of maximal entropy for the Teichmueller flow on our moduli space. The proof proceeds in Veech's space of zippered rectangles and involves approximation of the Teichmueller flow by a sequence of suspension flows over countable Bernoulli shifts with roof functions depending on only one coordinate. This method has been introduced by Gurevich in the '70's and developed by Gurevich and Savchenko in the '90's. The uniqueness of the measure of maximal entropy follows from a result of Buzzi and Sarig (2004).

27th November

** Christopher Jones (Warwick) **

**Homoclinic Bifurcation with Symmetry **

** **

4th December

** Manjula Samarasinghe (Queen Mary) **

**Quasi-fuchsian Correspondences **

11th December

** Todd Young (Ohio)**

**Topological entropies of equivalent smooth flows **

**Abstract:** Two flows defined on a smooth manifold are equivalent if there exists a homeomorphism of the manifold that sends each orbit of one flow onto an orbit of the other flow while preserving the time orientation. The topological entropy of a flow is defined as the entropy of its time-1 map. While topological entropy is an invariant for equivalent homeomorphisms, finite non-zero topological entropy for a flow cannot be an invariant because its value is affected by time reparameterization. However, 0 and $\infty$ topological entropy are invariants for equivalent flows without fixed points. In equivalent flows with fixed points there exists a counterexample, constructed by Ohno, showing that neither 0 nor $\infty$ topological entropy is preserved by equivalence. The two flows constructed by Ohno are suspensions of a transitive subshift and thus are not differentiable. Note that a differentiable flow on a compact manifold cannot have $\infty$ entropy. These facts led Ohno in 1980 to ask the following: "Is 0 topological entropy an invariant for equivalent differentiable flows?" In this paper, we construct two equivalent $C^\infty$ smooth flows with a singularity, one of which has positive topological entropy while the other has zero topological entropy. This gives a negative answer to Ohno's question in the class $C^\infty$.

This is joint work with Wenxiang Sun and Yunhua Zhou.

15th January

** Daniel Thompson (Warwick) **

**Topological pressure for non-compact sets **

**Abstract:** In the 80's, Pesin and Pitskel defined topological pressure for non-compact sets as a characteristic of dimension type, generalising a definition of topological entropy introduced by Bowen. We give an alternative definition of topological pressure in the non-compact setting via a suitable variational principle. We derive some properties of the new topological pressure and compare them with the properties of the Pesin and Pitskel pressure. We describe a simple example which illustrates the difference in the thermodynamic properties of the two quantities. We conclude with an example taken from the multifractal analysis of the Lyapunov exponent for the Manneville-Pomeau family of maps, which seems particularly well adapted to our new framework.

22nd January

**Dominique Fleischmann (Open University)**

** Dynamical properties of a family of entire functions.**

**Abstract:** In 1981 Noel Baker presented results concerning the types of components of the Fatou set of a transcendental function. He showed that if a transcendental entire function has an invariant Baker domain, that is, an invariant domain in which the iterates tend to infinity, then its growth must be at least order $1/2$ mean type. Baker demonstrated that this result is sharp by showing that the transcendental entire function $f_{c}$ defined by \begin{equation*} f_{c}(z) = z + {\sin \sqrt{z} \over \sqrt{z}} + c \end{equation*} has a Baker domain for sufficiently large positive $c$. In this talk we will examine the dynamical properties of this family of functions in more detail. In Dynamical properties of entire functions of small growthparticular, we will show how a new method can be used to prove that such functions have an invariant Baker domain for all $c > 0$. We will quantify the asymptotic distribution of singular values of the family, and compare this with a lower bound on the density of singular values of such functions that was proved by Detlev Bargmann in 2001. Lastly, we quantify the distribution and nature of the fixed points of the family and describe how the dynamics change as $c$ varies.

29th January

**Jeremy Kahn (Stony Brook University)**

** Progress towards the MLC conjecture. **

**Abstract:** A central conjecture of 1-dimensional holomorphic dynamics is that the Mandelbrot set is locally connected. This is equivalent to combinatorial equivalence implies conformal equivalence (for quadratic polynomials). By the work of Yoccoz (c. 1989) this implication remains to be shown only for infinitely renormalizable quadratic polynomials. We will review recent progress towards this conjecture, in the case where each intermediate renormalization has ''spinal entropy'' bounded below.

5th February

**Gwyneth Stallard (Open University)**

** Dynamical properties of entire functions of small growth **

Abstract: This talk concerns the iteration of transcendental entire functions - in particular, of such functions for which the growth is small and so, in some sense, the resemblance to a polynomial is as close as possible for a transcendental function. We show how the properties of such functions lead to a surprising link between two apparently unrelated conjectures and discuss recent progress on both conjectures. The first conjecture, due to Baker, concerns the components of the Fatou set (the set of points that are stable under iteration). The second conjecture, due to Eremenko, concerns the components of the escaping set (the set of points that tend to infinity under iteration). This is joint work with Phil Rippon.

12th February

**Ian Morris (Warwick)**

** Pythagorean triples and thermodynamic formalism. **

19th February

**Charles Pfister (École polytechnique fédérale de Lausanne)**

** On the topological entropy of saturated sets. **

**Abstract**: Let $(X,d,T)$ be a dynamical system, where $(X,d)$ is a compact metric space and $T:X\rightarrow X$ a continuous map. We introduce two conditions for the set of orbits, called respectively the g-almost product property and the uniform separation property. The g-almost product property holds for dynamical systems with the specification property, but also many others. For example, all $\beta$-shifts have the g-almost product property. The uniform separation property is true for expansive an more generally asymptotically h-expansive maps. Under these two conditions we compute the topological entropy of saturated sets. If the uniform separation property does not hold, then we can compute the topological entropy of the set of generic points, and show that for any invariant probability measure $\mu$, the (metric) entropy of $\mu$ is equal to the topological entropy of generic points of $\mu$. We give an application of these results to multi-fractal analysis and compare our results with those of Takens and Verbitskiy (Ergod. Th. & Dynam. Sys. 23 (2003), 317-348).

26th February

**John Roberts (University of New South Wales) **

**Periodic orbits of linear endomorphisms on the 2-torus and its lattices**

**Abstract**: Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local statistics and on what determines whether two different endomorphisms have the same statistics. The situation on the invariant rational lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant given in terms of the entries of the matrix associated with the toral endomorphism.

4th March 2008

**Agnieszka Badenska (Warsaw University of Technology/ Warwick)**

**Conformal and invariant measures for hyperbolic meromorphic functions - construction and some properties.**

**Abstract**: We consider hyperbolic transcendental functions (entire or meromorphic) of finite order satisfying so-called rapid derivative growth condition. We present results of V.Mayer and M.Urbanski concerning the construction of conformal and invariant probability measures for the considered functions and tame potentials, and also some their properties. Finally, we show further properties of the measures, such as real-analyticity of Jacobian of invariant measure (which leads to the measure rigidity in some classes of functions) and exponentially fast mixing on cylinders (when the dynamics is conjugate to the shift map).

11th March

** Edmund Harriss (Imperial College)**

** Rauzy Fractals from PV (Pisot) to Non-PV.**

**Abstract:** It is well known that Hyperbolic automorphisms of the 2-torus admit a markov partition. This markov partition is closely related to 1 dimensional tilings of the line called Sturmian tilings. In the general case any hyperbolic toral automorphism of the d-torus admits a Markov partition, however a result of Bowen shows that, if d > 2 and the automorphism is algebraically irreducible, the elements of the Markov partition can never have a smooth boundary: this boundary is a non-trivial fractal set, and is difficult to construct in the general case. In 1982 Gerd Rauzy found a method of constructing the markov partition of an irreducible Hyperbolic automorphisms of the 3-torus using a substitution rule on three letters. This work has been generalised to construct the Markov partition for many (conjecturally all) irreducible PV toral automorphisms. Recent work of Arnoux, Furukado, Ito and H has opened up this geometric construction in the more complicated non-PV case, where there is more than 1 eigenvalue of absolute value greater than 1. This talk will focus on the geometric construction of Rauzy fractals that give the Markov partitions and the related substitution tilings. Starting with the original example of Rauzy, and the general PV-case before showing how this construction can be generalised to the non-PV case.

22th April

** Renato Vitolo (Exeter)**

** The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: resonance `bubbles' and the Arnold resonance web.**

**Abstract:** Two model maps for the Hopf-saddle-node bifurcation of diffeomorphisms are constructed and examined. The maps are on the one hand near the flow of a `versal' HSN vector field unfolding, and, on the other hand, designed `as generic as possible' as diffeomorphisms go. The first part of the study is centered around a 1:5 resonance taking place on a frayed quasi-periodic Hopf bifurcation boundary in a suitable parameter plane. Several subordinate codimension 1 and 2 quasi-periodic bifurcations of invariant circles and two-tori occur in the neighborhood. Other phenomena of interest are the routes leading to strange attractors: in particular, a quasi-periodic period doubling route is presented. The model studied in the second part is constructed to describe the dynamics inside an attracting invariant two-torus, which arises through a quasi-periodic Hopf bifurcation. Resonances inside the two-torus yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps. Evidence is provided for the conjecture that heteroclinic intersections of the invariant manifolds of the saddle periodic points give rise to the occurrence of strange attractors contained in the two-torus: this yields a concrete route to the Newhouse-Ruelle-Takens scenario.

29th April

** No session**

6th May

** Mark Pollicott (Warwick)**

**Points of non differentiability for conjugacy maps.**

**Abstract:** Consider the simple example of two smooth expanding maps of the circle of the same degree. They are conjugated by a homeomorphism. Moreover, unless this conjugacy is also a smooth diffeomorphism, then it will be differentiable almost everywhere with zero derivative (with respect to Lebesgue measure). We address the question of how large is the set of points where the conjugacy is not differentiable. The same basic method applies in other settings (e.g., boundary correspondences for cocompact Fuchsian groups). This is joint work with Jordan, Kesseboehmer, and Stratmann.

13th May

** No session**

20th May

** Gabriel Paternain (Cambridge)**

**Transparent connections over negatively curved surfaces.**

**Abstract:** A unitary connection is said to be transparent if its parallel transport along every closed geodesic is the identity. We show that transparent connections over a closed negatively curved surface are locally unique up to gauge equivalence (global uniqueness fails). The proof is based on the non-abelian Livsic theorem and a suitable Fourier analysis on the unit tangent bundle of the surface.

**Special Session:** Wednesday, 28th May. Room MS04. 14:00-15:00.

** Anatoly Neishtadt (Loughborough)**

**On the adiabatic perturbation theory for systems with elastic reflections.**

**Abstract:** The adiabatic perturbation theory (APT) gives an approximate description of the dynamics of smooth Hamiltonian systems containing fast and slow variables. The APT procedure can also be formally employed for systems with elastic reflections. However, the validity of this formal approach does not follow from the existing results on the accuracy of APT for smooth system, because slow variables vary rapidly (instantaneously) at a reflection. We obtain estimates of the accuracy of APT for systems with elastic reflections for three well-known model problems: Fermi-Ulam model (a particle between slowly moving walls), rays in a smoothly irregular waveguide with reflecting walls, and an adiabatic piston. The advantage of the approach based on APT with respect to other known approaches is that it allows one to deal directly with an initial Hamiltonian system, rather than with the corresponding Poincar'e map, to consider systems with impacts in the same way as smooth systems, and simplifies calculations considerably. This is joint work with I.Gorelyshev.

3rd June

**Omri Sarig (Pennsylvania State University.)**

**The horocycle flow and the Laplacian on hyperbolic surfaces of infinite genus.**

**Abstract:** Horocycle flows on hyperbolic surfaces of infinite genus may admit a large class of (infinite) invariant Radon measures. M. Babillot has suggested that the abundance of measures is related to the abundance of positive eigenfunctions for the Laplace-Beltrami operator. A couple of years ago, F. Ledrappier and I have proved this for all regular covers of compact hyperbolic surfaces. Recently we have been able to extend this to a significantly larger class of surfaces, which we call "tame". These include, for example, all complete hyperbolic surfaces which can be obtained by gluing countably many pairs of pants whose boundary components have lengths less than a constant. This is joint work with F. Ledrappier.

10th June

**Anthony Manning (Warwick)**

**Curves of fixed points of trace maps.**

**Abstract:** We study certain diffeomorphisms of R^3 (where the coordinates are traces of 2x2 matrices). These diffeomorphisms preserve certain level surfaces where the maps are area-preserving with hyperbolic or elliptic behaviour. One such surface is a quotient of a torus and the action is by automorphisms. Hyperbolic fixed points persist at nearby levels and we explain in certain cases where such curves of fixed points return to that level. This work is joint with S. Humphries.

24th June

**Dragomir Saric (Queens College-CUNY)**

**Mapping class groups can not be realized by homeomorphisms. **