Institutionen för matematik, KTH | Matematiska institutionen, Uppsala

# DNA-seminariet (Dynamiska system, talteori, analys)

## våren 2009 (spring 2009)

#### 15 januari 2009, Olof Sisask (University of Cambridge): The minimal and maximal number of three-term progressions in dense subsets of Z/pZ

Abstract. A famous theorem of Roth asserts that any dense subset of the integers {1, ..., N} must contain a three-term arithmetic progression provided N is large enough in terms of the density of the set. This turns out to be equivalent to the statement that a subset of {1 ,..., N} of positive density delta must actually contain a lot of three-term progressions: at least c(delta)N^2 of them, in fact, where c(delta) is some positive constant depending only on the density delta. Similar statements exist in Z/pZ, the integers modulo a prime p, and I shall discuss the analogous problem in this setting: how many three-term progressions must A contain if A is a subset of Z/pZ of density delta? In particular, I shall outline how one can obtain an exact answer for very large densities using some analytically-inspired ideas. Based on joint work Ben Green.

#### 29 januari 2009, Pierre Berger (CRM, Barcelona): Abundance of one dimensional non uniformly hyperbolic attractors for surface endomorphisms

Abstract. We prove the existence of a non uniformly hyperbolic attractor for a positive set of parameters a in the family: $(x,y)\mapsto (x^2+a+2y,0)+B(x,y)$ Where $B$ is fixed a $C^2$ small function. The proof uses the formalism of Yoccoz puzzle and analytical ideas of Benedicks-Carleson.

#### 12 mars 2009, Thomas Kaijser (Linköpings universitet): Iterations of random functions. Contraction properties and limit theorems.

Abstract. (given as a pdf-file)

#### 16 mars 2009, Alexander Fish (Ohio State University): Sumset phenomenon for amenable groups

Abstract. We prove a structure theorem for a sumset of two sets A and B of positive upper Banach density in any countable amenable group. More precisely, we prove that AB is "piecewise syndetic" which means that there exists a finite set K such that for any finite set F in G ("configuration") there exists an element g in G such that Fg is a subset of ABK. For abelian groups we prove even more, namely, if A and B have positive upper Banach density then there exists a finite set K in G such that A+B+K is a piecewise Bohr set (large pieces of almost periodic set -- contains a lot of structure). The latter implies that there exist C, D, E sets of positive upper Banach density such that C+D+E is a subset of A+B. (joint work with M.Beiglbock and V.Bergelson)

#### 26 mars 2009, Jean-Pierre Conze (Institute of Mathematical Research of Rennes): Asymptotic laws for some sequential dynamical systems

Abstract. (given as a pdf-file)

#### 2 april 2009, Giorgos Costakis (University of Crete): Dynamics of linear operators in finite and infinite dimensions

Abstract. (given as a pdf-file)

#### 16 april 2009, Johan Andersson: Kloosterman sums and their applications in analytic number theory

Abstract. We will introduce the Kloosterman sums S(m,n;c) and discuss about some of their applications in analytic number theory, in particular applications on exponential sums and on the fourth power moment of the Riemann zeta-function (following Heath-Brown, Kuznetsov, Iwaniec and Motohashi).

#### 4 maj 2009, Björn Winckler (KTH), Renormalization fixed points: one algorithm to find them all

Abstract. In this talk I will give an overview of the renormalization theory for unimodal maps. The focus will be on Marco Martens' proof of the existence of renormalization fixed points and how it naturally leads to an algorithm for constructing such fixed points (of any combinatorial type and critical exponent). Finally, I will outline a computer implementation of this algorithm.

#### 7 maj 2009, Kristian Bjerkloev (KTH): Quasi-periodic perturbation of quadratic maps

Abstract. We consider quasi-periodic perturbations of a quadratic map exhibiting an attracting period-3 point. We will rigorously show that such a perturbation can create so-called Strange Nonchaotic Attractors, an object which lies between regularity and chaos.

#### 8 maj 2009, Michael Benedicks (KTH): Kneading sequences for the Double Standard Map

Abstract. Maps from double standard map family f_a(x)=2x+a+(1/pi) sin(2 pi x) (mod~1), have the property that they are double covers of the circle onto itself with a unique inflexion point. They have been investigated most recently by M. Misiurewicz and A. Rodrigues. In particular one can say that they are hybrids between circle homeomorphims with inflexions and quadratic maps of the interval. The aim of the talk is to develop symbolic dynamics and kneading theory for these maps and discuss the behaviour in parameter space (chaotic behaviour, stable periodic orbits) comparing the situation to the more standard cases of circle homeomorphisms and quadratic interval maps. This is joint work with A. Rodrigues.

#### 14 maj 2009, Elena Ushakova (UU): Kernel operators with variable limits of integration in Lebesgue spaces

Abstract. We study L^p-L^q boundedness and compactness of the operator f -> w(x) int_{a(x)}^{b(x)}k(x,y)f(y)v(y)dy with given weight functions w(x),v(y), differentiable strictly increasing border functions a(x),b(x) and a kernel k(x,y) satisfying some growth conditions. The results are applied for weighted L^p-L^q boundedness of geometric mean operator f -> exp[(b(x)-a(x))^(-1) int_a(x)^b(x) log f(y) dy] and other related problems. The talk is based on U.U.D.M. Reports 2008:30 and 2008:46.

Abstract. This talk will present new developments in understanding the analytic continuation of certain Dirichlet series in several complex variables associated to moments of quadratic Dirichlet L-functions.

#### 25 maj 2009, Emmanuel Breuillard (Univerité Paris-Sud, Orsay): Equidistribution of dense subgroups of nilpotent Lie groups

Abstract. The question of equidistribution of Gamma orbits on a homogeneous space X has been thoroughly studied in recent years from many perspectives. In this talk I will tackle this question for Gamma a nilpotent group and X a nilpotent Lie group and consider two types of averages: the word length average and the random walk average. Using unique ergodicity and precise geometric information on the shape of nilpotent balls I will show how to answer the equidistribution problem in that setting.

#### 25 maj 2009, Uri Shapira (Hebrew University): Applying dynamics to number theory.

Abstract. I will present a recent joint work with Manfred Einsiedler and Lior Fishman in which we use rigidity results in dynamics to prove results in Diophantine approximations. We study how certain fractals intersect certain Diophantine classes. In particular I plan to concentrate on the following theorem regarding the intersection of the middle third Cantor set and the set of "Well Approximable" numbers: Theorem: Let a_n be a random sequence of the digits 0 and 2 (each digit appears with probability 1/2) and let x be the number in the unit interval having this sequence as its base three expansion. Then with probability one the coefficients in the continued fraction expantion of x, are unbounded.

#### 25 maj 2009, Ben McReynolds (University of Chicago): Geometric spectra.

Abstract. In this talk, I will give a brief review of classical spectral geometry and the study of the geodesic length spectrum on a Riemannian manifold. I will then discuss some generalizations of the length spectrum and some results on how much of the geometry is encoded in other geometric spectra. This is joint work with Alan Reid.

#### 2 juni 2009, Tom Sanders (Inst Mittag-Leffler): Modeling Roth's theorem on three term arithmetic progressions

Abstract. A beautiful theorem of K. F. Roth from the 50's asserts that any subset of the integers containing no three-term arithmetic progressions with non-zero common difference has density zero. In the 80's and 90's a beautiful model problem was considered: suppose that A \subset (\Z/3\Z)^n contains no affine line. Then |A|=O(3^n/n). A proof of this result (due to Meshulam) can be seen as a finite field version of Roth's proof of his aforementioned theorem, and in this setting the argument becomes much simpler. Despite this no improvement is known and any bound of the shape o(3^n/n) would be of considerable interest. In this talk we shall consider the analogous problem for (\Z/4\Z)^n where an improvement over Roth's argument is possible.

#### 9 juli 2009, Carlos Vasquez: Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents.

Abstract. In this talk, we establish stable ergodicity for diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the center direction are all positive with respect to the physical measures.

#### 9 juli 2009, Thomas Tucker (University of Rochester): Dynamical Mordell-Lang problems

Abstract. Let S be a group or semigroup acting on a variety V, let x be a point on V, and let W be a subvariety of V. What can be said about the structure of the intersection of the S-orbit of x with W? Does it have the structure of a union of cosets of subgroups of S? The Mordell-Lang theorem of Laurent, Faltings, and Vojta shows that this is the case for certain groups of translations (the Mordell conjecture is a consequence of this). On the other hand, Pell's equation shows that it is not true for additive translations of the Cartesian plane. We will see that this question relates to issues in complex dynamics, simple questions from linear algebra, and techniques from the study of linear recurrence sequences.