ERGODIC THEORY AND STATISTICAL MECHANICS SEMINAR Thursday 2.00 PM, Room 801, Fine Hall, Princeton University. For informations: fcellaro AT math DOT princeton DOT edu *** THE ROOM HAS CHANGED: NOW WE MEET IN FINE 801 *** March 3rd 2011 Ivan Corwin (Courant Institute, NYU) Title: Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions Abstract: We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions -- the crossover distributions -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes. Friday, March 11th 2011 ***Notice special day!*** Nikolai Chernov (University of Alabama at Birmingham) Title: Diffusion in a periodic Lorentz gas with narrow tunnels (P. Balint, N. Chernov, and D. Dolgopyat) Abstract: In a periodic Lorentz gas a particle moves bouncing off a regular array of fixed convex obstacles (scatterers), like in a pinball machine. When the horizon is finite, one observes a classical diffusion law. When the obstacles are so large that the tunnels between them become narrow (of width $\epsilon \to 0$) then the diffusion matrix scales with $\epsilon$. In the limit, when $\epsilon=0$, the particle is trapped in a compact region with cusps in the boundary. In that case the system ceases to be uniformly hyperbolic and develops anomalies. Correlations decay slowly, as $1/n$, and the classical central limit theorem fails. Instead, a non-classical limit law holds, with a scaling factor of $\sqrt{n\log n}$ replacing the standard $\sqrt{n}$. However, for a special observables whose average values at the cusps vanish, the classical central limit law still holds. *** There will be no seminar on March 17th *** March 24th 2011 Corinna Ulcigrai (University of Bristol) Title: TBA Abstract: TBA March 31st 2011 TBA (TBA) Title: TBA Abstract: TBA April 7th 2011 TBA (TBA) Title: TBA Abstract: TBA April 14th 2011 TBA (TBA) Title: TBA Abstract: TBA April 21st 2011 Alexander Bufetov (Rice University) Title: TBA Abstract: TBA April 28th 2011 TBA (TBA) Title: TBA Abstract: TBA PAST SEMINARS: February 17th 2011 Han Li (Yale) Title: Effective Limit Distribution of the Frobenius Numbers Abstract: The Frobenius number of a lattice point $\bf{a}$ with positive coprime coordinates, is the largest integer which can NOT be expressed as a non-negative integer linear combination of the coordinates of $\bf{a}$. Marklof showed in 2010 that the limit distribution of the Frobenius numbers is given by the distribution for the covering radius function of a random unimodular lattice. The aim of the talk is to discuss the reason of this phenomenon, and indicate how to obtain the rate of the convergence of the corresponding distribution functions. February 10th 2011 Mira Shamis (IAS) Title: Discrete Schrodinger operators with periodic and almost periodic potentials Abstract: The first half of the talk will be a survey; I will start from general facts about discrete Schrodinger operators with periodic potentials, and then discuss operators with almost periodic potentials, focusing on the almost Mathieu operator. In the second half, I will state and prove some results connecting the absolutely continuous spectrum of almost periodic operators to the spectra of their periodic approximations. December 2th 2010 Zhiren Wang (Princeton) Title: Two results on rigidity of commutative actions by toral automorphisms Abstract: In 1983 Berend proved rigidity of higher-rank commutative actions by toral automorphisms under some hyperbolicity and irreduciblity assumptions. We will present two rigidity results that respectively extend Berend's theorem to certain non-hyperbolic and reducible cases. We will also discuss some counterexamples of non-homogeneous orbit closures. This is joint work with Elon Lindenstrauss. November 11th 2010, November 18th 2010 Percy Wong (Princeton) Title: A survey of results in universality of Wigner matrices Abstract: In the 1950's, Wigner proved the famous semicircle laws for Wigner matrices and started the study of universality results in random matrices.  In these two talks, this will serve as our starting point as we surveyed the historical developments in this field.  We will end with a discussion of the proof of the local semicircle law of Erdos, Schlein and Yau and the four moment theorem by Tao and Vu.  We will also discuss some of the open problems in the study of random matrices if time permits. October 29th 2010 Domokos Szász (Budapest University of Technology) Title: Diffusive or superdiffusive asymptotics for periodic and non-periodic Lorentz processes Abstract: After the first success in establishing the diffusive, Brownian limit of planar, finite-horizon, periodic Lorentz processes, in 1981 Sinai turned the interest toward studying models when periodicity is hurt, in particular, to locally perturbed Lorentz processes. The 1981 solution for a random-walk-model - by Telcs and the speaker - only led in 2009 to that for the locally perturbed, finite-horizon Lorentz process (by Dolgopyat, Varjú and the present author). Beside reporting on these results we also analyze the first steps in extending the limits obtained for the periodic Lorentz process to locally perturbed periodic or quasi-periodic ones (results by Nándori, Paulin, Varjú and the speaker). October 21st 2010 Vadim Gorin (Moscow State University) Title: From random tilings to representation theory Abstract: Lozenge tilings of planar domains provide a simple, yet sophisticated model of random surfaces. Asymptotic behavior of such models has been extensively studied in recent years. We will start from recent results about q-distributions on tilings of a hexagon or, equivalently, on boxed plane partitions. (This part is based on the joint work with A.Borodin and E.Rains). In the second part of the talk we will explain how representation theory of the infinite-dimensional unitary group is related to random lozenge tilings with a certain Gibbs property. We will discuss applications of this correspondence and results on the classification of Gibbs measures on tilings of the half-plane. October 14th 2010 Amit Singer (Princeton) and Xiuyuan Cheng (Princeton) Title: The Spectrum of an Hermitian Matrix With Dependent Entries Constructed from Random Independent Images Abstract: In this talk we will present a preliminary analysis and numerical results for the distribution of eigenvalues of a certain random N by N  Hermitian matrix, whose construction is motivated by a problem in structural biology. The matrix is built from N images, where each image is an array of P pixels, and the pixels are i.i.d standard Gaussians. Numerical experiments suggest that the spectrum approaches Wigner's semi-circle law for P>>N, but differs significantly from the semi-circle for P<<N. We attribute this difference to the dependencies among the matrix entries. In particular, using the third moment, we show that for fixed P, the largest eigenvalue is O(N) as N goes to infinity. October 7th 2010 John Mather (Princeton) Title: Shortest Curves Associated to a Degenerate Jacobi Metric on the two Torus. Abstract:  Let P be a potential on the two torus that takes its minimum value at a unique point m.  Set E_0 := P(m).  For a real number E, let g_E be the Jacobi metric associated to P and E.  For E > E_0, g_E is a Riemannian metric.  An ancient theorem of Morse and Hedlund says that a g_E-shortest curve in an indivisible homology class is simple.  For E = E_0, g_E is no longer a Riemannian metric because it vanishes at m.  (It is a Riemannian metric in the complement of m.)  For a suitable potential P, and a suitable indivisible homology class h, a g_E_0-shortest curve in h crosses itself at m, so the theorem of Hedlund and Morse does not generalize to the case E = E_0.  In this talk, I will describe examples of such shortest curves that cross themselves and give a few ideas of how to prove that each such shortest curve does not cross itself except at m and is a bouquet of simple curves in at most three homology classes. September 17th 2010 Tadashi Tokieda (University of Cambridge, UK) Title: Dynamics of 2D point vortices and its generalizations. April 30th 2010 Dmitry Dolgopyat (University of Maryland) Title: Dynamics of bouncing balls. Abstract: We consider a ball bouncing off infinitely heavy periodically moving wall in the presence of a potential force. We are interested in the question how large is the set of orbits whose energy tends to infinity. Both smooth and piecewise smooth motions of wall will be considered. We also present some related questions about small piecewise smooth perturbations of nearly integrable systems. April 15th 2010 Alex Kontorovich (Brown University and IAS) Title: Applications of Expansion and Equidistribution to Number Theory Abstract: We will discuss recent applications of Expander Graphs to various problems in Equidistribution and Sieving. April 8th 2010 Jean-René Chazottes (CNRS and École-Polytechnique) Title: Concentration inequalities for dynamical systems Abstract: concentration inequalities are a powerful tool to estimate the fluctuations of observables more general than ergodic sums: one can consider any observable F(x,...,T^n x) provided it is separately Lipschitz. Such inequalities can be established for non-uniformly hyperbolic systems and we shall present some applications. April 1st 2010 Vitaly Bergelson (Ohio State University) Title: Independence properties of weakly mixing systems and polynomial patterns in large sets Abstract: Various recurrence and convergence results obtained in recent years indicate that dynamical systems exhibit regular behavior along polynomial times. In particular, weakly mixing systems turn out to always possess rather strong independence properties along certain sets of zero density. We will discuss some implications of these results in physics as well as applications to combinatorics and number theory (including polynomial extensions of Szemeredi's theorem on arithmetic progressions and recent work of Tao and Ziegler on polynomial patterns in primes). We will also formulate and discuss some natural open problems and conjectures. March 25th 2010 Ilya Vinogradov (Princeton) Title: Gaps in n^(1/2) mod 1 and Ergodic Theory Abstract: In 2004 Elkies and McMullen proved that gaps for n^(1/2) mod 1 have a limiting distribution and computed its density. The first ingredient in the proof is an equidistribution theorem for particular horocycle lifts to a bundle over the modular surface. The second is a linearization of the counting function that is special to the power 1/2. I will present these two pieces of the proof and show how they produce a limiting distribution. March 11th 2010 John Pardon (Princeton) Title: Random polygons in plane convex sets Abstract:  Consider picking N random points in a convex set K and forming their convex hull K_N.  Recently, there have been a number of results concerning the asymptotic behavior of random variables such as the area and number of vertices of K_N.  These are, however, all limited to two special cases: 1) K is a polygon and 2) K is "smooth".  I will discuss work which obtains uniform bounds over the family of all convex sets K.  These results include central limit theorems for the area and number of vertices of K_N. March 5th 2010, 2.30pm Pablo Shmerkin (University of Manchester) Title: The dimension of self-affine sets: past, present and future. Abstract: Calculating the dimension of sets invariant under non-conformal dynamics is a formidable problem. My talk will be a survey on what is known and expected for self-affine sets, i.e. sets invariant under piece-wise affine expanding maps on Euclidean space. Some emphasis will be given to my joint work with A. Käenmäki on self-affine sets of Kakeya type, and the thermodynamic formalism for the singular value function. March 4th 2010 Francesco Cellarosi (Princeton) Title: On the limit curlicue process for theta sums Abstract: I shall discuss a random process achieved as the limit for the ensemble of curves generated by interpolating the values of theta sums. The existence and the properties of this process are established by means of purely dynamical tools and rely on generalizations of a result by Marklof and Jurkat and van Horne. Joint work with Jens Marklof. February 25th 2010 Vladislav Vysotsky (University of Delaware) Title: Limit theorems for sticky particle systems and positivity of integrated random walks Abstract: Consider the model of a one-dimensional gas, whose particles have random initial positions and random initial velocities. Particle attract each other due to gravitation, and stick together at collisions. As time goes, the number of particles decreases while their sizes increase until there forms a giant single particle of the total mass. The results on this process of mass aggregation are given in form of limit theorems as the number of initial particles tends to infinity. For example, the stochastic processes of the total number of particles satisfy a functional central limit theorem. I will show how this problem on the number of particles is related to positivity of integrated random walks. I will discuss the problem of finding one-sided small deviation probabilities of integrated random walks and other stochastic processes, and tell about the progress in this field. February 18th 2010 Zeev Rudnick (Tel-Aviv University and IAS) Title: Nodal sets for eigenfunctions of the Laplacian and lattice points on circles and spheres. Abstract: An outstanding problem in "Quantum Chaos" is to understand the complexity of highly excited quantum states, and in particular of eigenfunctions of the Laplacian on a compact manifold. One attribute that can be studied is complexity of the nodal set, which is the locus of points where the eigenfunction vanishes. I will describe some recent results for the case of flat tori, concerning  the structure of the nodal sets and L2-bounds for the restriction of the eigenfunctions to hypersurfaces. These issues connect with purely arithmetic problems on the representations of integers as sums of squares and the distribution of the corresponding lattice points on arcs of circles and caps on spheres. (Joint work with Jean Bourgain) February 11th 2010 Artur Avila (CNRS and IMPA) Title: Convergence of renormalizations. February 4th 2010 Artur Avila (CNRS and IMPA) Title: The spectral dichotomy for one-frequency Schrodinger operators. Abstract: In the theory of one-frequency Schrodinger operators, the best understood potentials have been those that can be somehow considered either small or large.  Roughly, small potentials tend to inherit the behavior of the Laplacian and present absolutely continuous spectral measures (leading to good transport properties), while for large potentials it is Anderson localization that prevails. Dynamically, those two distinct local theories correspond to a good understanding of cocycles near constant (“the domain of KAM”), and nonuniformly hyperbolic cocycles.  We have proposed to build a global theory by focusing on the description of the phase-transition, from KAM to non-uniform hyperbolicity, in the infinite-dimensional space of cocycles.  Its goal was to prove the Spectral Dichotomy: typically, an operator is the direct sum of a “small-like” and a “big-like” operators, with disjoint spectra. We will describe the structure of the proof of the Spectral Dichotomy, which has two main parts.  The firt one describes the locus of criticality from the point of view of Lyapunov exponents, i.e., the boundary of nonuniform hyperbolicity.  The second one relates zero Lyapunov exponents and KAM. December 11th 2009, 2pm, Room 401 Michael Boshernitzan (Rice University) Title: Diophantine Properties of Dynamical Systems and IETs. Abstract: The lecture is based on a recent preprint with the same title, joint with J. Chaika and put recently on arXiv.  One of the results is that for ergodic IETs (Interval Exchange Transformations) almost sure $\liminf_{n\to\infty}\limits n|T^nx-y|=0$. The result is optimal in two ways:  (1) the normalizing factor \ $n$ \  cannot be improved, even for rotations; (2) the assumption of ergodicity cannot be replaced by just minimality. Several open problem (related) will be presented. December 3rd 2009 Boris Rozovsky (Lefschetz Center for Dynamical Systems, Brown University) Title: Unbiased Random Perturbations of Navier-Stokes Equation Abstract: A random perturbation of a deterministic Navier-Stokes equation is considered in the form of an Stochastic PDE with Wick product in the nonlinear term. The equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. The generalized solution is obtained as an inverse of solutions to corresponding quantized equations. An interesting feature of this type of perturbation is that it preserves the mean dynamics: the expectation of the solution of the perturbed equation solves the underlying deterministic Navier-Stokes equation. From the stand point of a statistician it means that the perturbed model is unbiased. The talk is based on a joint work with R. Mikulevicius. November 20th 2009, 3pm, Room 401 Elon Lindenstrauss (Princeton) Title: A brief survey of effective equidistribution results in Gamma\G Abstract: Equidistribution results for orbits and more general configurations in Gamma\G are a central focus of the theory of flows on homogeneous spaces. A notable example that comes to mind is Ratner's equidistribution theorem. I will survey some old and new quantitive equidistribution results of this flavor by several authors. November 19th 2009 Lai-Sang Young (Courant Institute NYU) Title: Ergodicity of some boundary driven integrable Hamiltonian chains Abstract: Small Hamiltonian systems are connected in a chain the ends of which are coupled to unequal heat baths, forcing the system out of equilibrium. Energy exchange is of a form that leads to integrable dynamics. A proof of ergodicity of both equilibrium and nonequilibrium steady states will be presented. This is followed by numerical results which show that unlike certain mechanical systems with chaotic microdynamics, marginal distributions of NESS in these integrable chains are not Gibbsian, leading to problems in the definition of “local temperature”. October 22nd 2009 Mikko Stenlund (Courant Institute NYU) Title: A pair correlation bound implies the Central Limit Theorem for Sinai Billiards Abstract: It is an open problem in the study of dynamical systems whether fast decay of correlations alone is sufficient for the Central Limit Theorem (CLT) to hold. On the one hand, there are no examples of dynamical systems for which correlations decay quickly but the CLT fails. On the other, existing CLT proofs rely on statistical properties much stronger than correlation decay. In the talk I will discuss a prime class of physically relevant systems, called Sinai Billiards, and show that a single  bound on correlations indeed implies the CLT directly. As a byproduct, the CLT is obtained for observables possessing remarkably little regularity October 15th 2009 Jeremy Kahn (SUNY Stony Brook) Title: A priori bounds for bounded-primitive renormalization Abstract: We say that an infinitely renormalizable quadratic polynomial has bounded-primitive type if we can find an infinite sequence of primitive renormalization times, such that the ratio between consecutive terms of the sequence is bounded. We prove that any such polynomial has the a priori bounds: there is a lower bound on the modulus of all renormalizations. This implies that the Mandelbrot set is locally connected at the associated parameter values. October 8th 2009 Ilya Vinogradov (Princeton University) Title: Nearest neighbor distances for several rotations Abstract: We will discuss results of Marklof on distributions of nearest neighbor distances. We will look at the Poisson scaling and at CLT scaling. Another point of view is to look at the number of distinct gap lengths in this scenario. Here we will explain unpublished results of Boshernitzan and Dyson. October 1st 2009 Yakov G. Sinai (Princeton University) Title: The decay of Fourier modes in solutions of Navier-Stokes systems February 12th 2009 Joint talk by: Yakov G. Sinai (Princeton University) Ilya Vinogradov (Princeton University) Title: Limiting Distribution of Large Frobenius Numbers. February 19th 2009 Joint talk by: Dong Li (Institute for Advanced Study) Title: The decay of Fourier modes of solutions of 2-D Navier-Stokes system Nikolai I. Chernov (University of Alabama) Title: Numerical results related to the talk by D. Li. February 26th 2009 - 4pm Dmitry Ostrovsky Title: Limit lognormal process, Selberg integral as Mellin transform, and intermittency differentiation. Abstract: The limit lognormal process is a multifractal stochastic process with the remarkable property that its positive integral moments are given by the celebrated Selberg integral. We will give an overview of the limit lognormal construction followed by a summary of our results on functional Feynman-Kac equations and resulting intermittency expansions that govern its distribution. The talk will focus on the intermittency expansion for the Mellin transform. This expansion recovers Selberg’s formula for the positive integral moments and gives a novel product formula for the negative ones. By summing it in general using a moment constant method, we obtain an extension of Selberg’s finite product to the Mellin transform of a probability distribution in the form of an infinite product of ratios of gamma functions in the complex plane. This distribution is conjectured to be the limit lognormal distribution. March 3rd Jonathan Mattingly (Duke University) Title: What makes the ergodic theory of Markov chains in infinite dimensions different (and difficult) ? Abstract: I will discuss how Markov chains in infinite dimensions generically have typically have properties  which  make their ergodic theory difficult. Such properties are very pathological  in finite dimensions, but in some sense generic in infinite dimensions. I will draw examples from stochastically forced PDEs and stochastic delay equations. We will see that in infinite dimensions, a typical system acts much more like an hypo-elliptic diffusion then an elliptic one. I will also discuss the existence of spectral gaps as well as the uniqueness of invariant measures. If time permits I will discuss an extension of Hormander's "sum of squares theorem" to infinite dimensions. March 5th 2009 Marco Martens (State University of New York at Stony Brook) Title: Hénon Renormalization Abstract: The geometry of strongly dissipative infinite renormalizable Hénon maps of period doubling type is surprisingly different from its one-dimensional  counterpart. There are universal geometrical properties. However, the Cantor attractor is not geometrically rigid. Typically, it doesn't have bounded geometry. The average Jacobian is a topological invariant of the global attractor. Although the geometry of the Cantor attractor can be deformed by changing the average Jacobian, the geometry is universal in a distributional sense. March 12th 2009 Thomas Spencer (Institute for Advanced Study) Title: Random walks with memory and statistical mechanics. Abstract: This talk will review some results and conjectures about history dependent random walks. For example, edge reinforced random walk (ERRW) is a random walk which prefers to visit edges it has visited in the past. Diaconis showed that ERRW can be expressed as a random walk in a random environment. This environment is highly correlated and is described in terms of statistical mechanics.  Phase transitions for closely related models are believed to occur in three dimensions.  One phase corresponds to diffusion and the other phase to localization. This talk is based work of Merkl and Rolles on ERRW and my recent preprint with Disertori and Zirnbauer on a hyperbolic sigma model. March 26th 2009 Manfred Denker (Penn State University) Title: Local limit theorems in ergodic theory Abstract: We use Stone's version of a local limit theorem from 1969: Let $(X,{\cal F},T,m)$ be a measure preserving dynamical system. A measurable  function $f:X\to \mathbb R$ satisfies a local limit theorem, if there are constants $A_n$ and $B_n\to\infty$ such that $$B_nm( f+f\circ T+...+f\circ T^{n-1} \in x_n+I) \to g(x)|I|,$$ where $(x_n-A_n)/B_n \to x$ and where $g$ is the density of some stable distribution. An analogous definition applies in the lattice case. For Gibbs-Markov dynamical systems (including certain Markov shifts), such results can be established when the function is in the domain of attraction of a stable distribution.  It also generalizes to non-Markov situations for certain maps of the interval, including beta-transformations. Applications to conservativity of dynamical systems and to Poincaré exponents are briefly discussed. April 2nd 2009 Thierry Bodineau (Institute for Advanced Study) Title: Large deviations of the current and phase transitions Abstract: Using the framework of the hydrodynamic limits, we will discuss the large deviations of a particle current through a diffusive system. The deviations can lead to dynamical phase transitions. In the case of asymmetric dynamics we will explain how the large deviation functional of the current provides a physical interpretation to the non-entropic solutions of Burgers equation. April 9th 2009 Corinna Ulcigrai (University of Bristol) Title: On mixing properties of locally Hamiltonian flows on surfaces Abstract: We consider area-preserving flows on surfaces which are locally given by smooth Hamiltonians. It turns out that the presence or absence of mixing depends on the type of fixed points. We proved in our PhD thesis that the presence of centers is generically enough to create mixing. Recently we showed that if such flows have only saddles, they are generically not mixing, but weakly mixing. The results use the flows representation as suspensions over interval exchange transformations and the study of deviations of Birkhoff averages over interval exchanges. April 16th 2009 Giovanni Forni (University of Maryland) Title: Existence/Non-existence of limiting distributions for horocycle flows on compact surfaces of constant negative curvature Abstract: A few years ago we have proved in collaboration with L. Flaminio that some non-trivial limit distributions for the horocycle flow must have compact support. In this talk we will refine that result and describe an existence/non-existence result for limiting distributions. In fact it turns out that whether limiting distributions exist or not depends on the geometry of the surface (via the eigenvalues of the Laplace operator) and on the observable under consideration. The main new idea is to express the precise results on the asymptotics of ergodic averages for the horocycle flow in terms of a dynamically defined cocycle which has the correct scaling property under the dynamics of the geodesic flow. Such cocycles are closely related to the invariant distributions of Flaminio-Forni and are analogous to the coycles constructed by A. Bufetov for asymptotic foliations of a Markov compactum (and in particular for area-preserving flows on higher genus surfaces). This is joint work with A. Bufetov. April 21st 2009 - 4.30 PM Ilya Goldsheid (Queen Mary, University of London) Title: An explicit approach to the control of Lyapunov exponents. Abstract: I shall discuss a new approach to the proof the exponential growth of products of random matrices. The classical Furstenberg's analysis relies on properties of infinite-dimensional unitary representations. The method I am going to discuss uses finite-dimensional representations and allows one to have a more explicit control over Lyapunov exponents. April 23rd 2009 Michael Hochman (Princeton University) Title: Local entropy and projections of dynamically defined fractals Abstract: If a closed subset X of the plane is projected orthogonally onto a line, then the Hausdorff dimension of the image is no larger than the dimension of X (since the projection is Lipschitz), and also no larger than 1 (since it is a subset of a line). A classical theorem of Marstrand says that for any such X, the projection onto almost every line has the maximal possible dimension given these constraints, i.e. is equal to min(1,dim(X)). In general, there can be uncountably many exceptional directions. An old conjecture of Furstenberg is that if A, B are subsets of [0,1] invariant respectively under x2 and x3 mod 1, then for their product, X=AxB, the only exceptional directions in Marstrand's theorem are the two trivial ones, namely the projections onto the x and y axes. Recently, Y. Peres and P. Shmerkin proved that this is true for certain self-similar fractals, such as regular Cantor sets. I will discuss the proof of the general case, which relies on a method for computing dimension using local entropy estimates. I will also describe some other applications. This is joint work with Pablo Shmerkin. April 30th 2009 Mikhail Lyubich (State University of New York at Stony Brook) Title: Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics Abstract: In a classical work of 1950's, Lee and Yang proved that zeros of the partition functions of the Ising models on graphs always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for a special ``Diamond Hierarchical Lattice". In this case, it can be described in terms of the dynamics of an explicit rational map in two variables. We prove partial hyperbolicity of this map on an invariant cylinder, and derive from it that the Lee-Yang zeros are organized asymptotically in a transverse measure for the central foliation. From the global complex point of view, the zero distributions get interpreted as slices of the Green (1,1)-current on the projective space. It is a joint work with Pavel Bleher and Roland Roeder.