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Furstenberg Conference Abstracts

Andrey Gogolev, "Moduli of smooth conjugacy of Anosov diffeomorphisms of Tⁿ, n>2": Structural stability asserts that if two Anosov diffeomorphisms are close enough then they are conjugate: hf=gh. It's known that the conjugacy h is Hölder continuous. There are simple obstructions for h to be smooth. Let p be a periodic point of f, fⁿ(p)=p, g^n(h(p))=h(p). If h were differentiable, then the differentials D(fⁿ)(p) and D(gⁿ)(h(p)) would be conjugate by the differential of h, and we say that f and g have same periodic data. De la Llave showed that if f and g are Anosov diffeomorphisms of T², then conjugacy is smooth if f and g have the same periodic data. For diffeomorphisms of Tⁿ, n>3, the corresponding statement doesn't hold -- periodic data is not a complete set of invariants for smooth conjugacy.
We consider Anosov diffeomorphisms of Tⁿ, n>2. Under some additional assumptions we prove that the conjugacy h is smooth if periodic data of f and g coincide. In case n=3 we require that the Mather spectra of f and g have 3 connected components -- f and g can be viewed as partially hyperbolic diffeomorphisms. This is joint work with Misha Guysinsky.

Neil Hindman, "Largeness of the set of finite products in a semigroup":

Elon Lindenstrauss, "Periodic orbits on the space of lattices": We consider periodic orbits of the full diagonal group on the space of lattices, SL(n, R)/SL(n, Z). If n=2 this amounts to considering periodic geodesics on the modular surface, and these are well known to behave in almost arbitrary way. However, a theorem of Duke states that if one considers all periodic orbits of a specific length these collections (or *packets*) become equidistributed. We give the analogous statement for n=3 using a combination of tools from automorphic forms and ergodic theory. This is joint work with Einsiedler, Michel and Venkatesh.

Alex Lubotzky, "Finite groups and hyperbolic manifolds": The isometry group of a closed hyperbolic n-manifold is finite. In 1974, Leon Greenberg proved the converse for n=2, i.e., every finite group is the full isometry group of some closed hyperbolic surface. Kojima (1988) extended the result to n=3. We will show the general case: For every n > 1 and every finite group G there is an n-dimensional closed hyperbolic manifold whose isometry group is isomorphic to G. An interesting aspect of the proof is that it is nonconstructive; it uses a 'probabilistic method', i.e., counting results from the theory of 'subgroup growth'. The talk won't assume any prior knowledge on the subject, and is based on joint work with M. Belolipetsky (Invent. Math. Dec. 2005)

Gregory Margulis, "Furstenberg boundary theory and rigidity": The purpose of the talk is to explain the role which Furstenberg boundary theory played in the proof of various rigidity results.

Larry Shepp, "A new look at Radon inversion: Representing a function of two variables as a superposition over angles of ridge functions with a given shape": This is joint work with Robert Perlmutter, Varian Semiconductor Equipment Corp. A ridge function with shape function g in the vertical direction is a function of the form h(x, 0) g(y). Along each vertical line it has the shape g(y), multiplied by a function h(x, 0) which depends on the x-value of the vertical line. Similarly a ridge function with shape function g in the horizontal direction has the form g(x) h(y, pi/2). For a given shape function g it may or may not be possible to represent an arbitrary function f(x,y) as a superposition over all angles of a ridge function with shape g in each direction, where h depends on the direction. We show that if g is Gaussian centered at zero then this is always possible and we give the function h for a given f(x,y). For highpass shapes g, however, we show it is impossible to represent an arbitrary f(x,y).

Christian Skau, "Orbit equivalence for Cantor minimal Zⁿ systems": We study actions of the free abelian group on n generators, acting as homeomorphisms on the Cantor set such that all orbits are dense. Our goal is to show that these systems are (topologically) orbit equivalent to Cantor minimal systems, i.e., to minimal Z-actions. We can prove this for n=2, and we will also sketch the strategy to prove this for any n. A crucial ingredient in such a proof is the so-called Absorption Theorem, which is a result about extending an AF-equivalence relation such that the new relation is orbit equivalent to the original one.

Arkady Tempelman, "Pointwise ergodic theorems for group representations in L^p":

Jim Tseng, "The Shrinking Target Properties of Circle Rotations": We give a necessary and sufficient condition for a circle rotation to have the s-exponent monotone shrinking target property (sMSTP), and, thereby we generalize a result for s = 1 that was established by J. Kurzweil and rediscovered by B. Fayad. If time permits, we will apply our technique to give a new, very short, proof of the logarithm law for irrational rotations.

Tamar Ziegler, "Polynomial progressions in primes": I will describe recent work with T. Tao in which we establish the existence of arbitrarily long polynomial progressions in primes. We prove a "transference" principle, similar to the one proved by Green and Tao, which enables us to reduce the problem of finding polynomial progressions in primes to a similar problem in sets of positive upper density in the integers. We then use a quantitative version of the Bergelson-Leibman polynomial Szemeredi theorem.