Manchester building, Hebrew University of Jerusalem, (Room 209)
A minimal representative for a dynamical system is a system that has the simplest possible dynamics in its topological equivalence class. This is very much related to "dynamical forcing": when existence of certain periodic orbits forces existence of others. This is quite useful in the analysis of chaotic systems. I'll give examples of minimal representatives
Manchester building, Hebrew University of Jerusalem, (Room 209)
We consider Bourgain's sparse ergodic theorem for systems where quantitative mixing estimates are present. Focusing on the case of the horocyclic flow, we show how to use such
estimates in order to bound the dimension of the exceptional set, providing evidence towards conjectures by N. Shah, G. Margulis and P. Sarnak. Moreover we show that there exists a bound which is independent from the spectral gap. The proof uses techniques from homogeneous dynamics, automorphic representations and number theory.
Consider a process in which points are assigned uniformly and independently at random on the interval [0,1]. It is a classical observation that after N points were assigned, the typical discrepancy of the empirical distribution, i.e., the maximum difference between the proportion of points on any interval and the length of that interval, is of order 1/sqrt{N}. Now consider a similar online process in which at every step an overseer is allowed to choose between two independent, uniformly chosen points on [0,1].
In ergodic dynamical systems almost every point is generic. Many times it is interesting to understand how large is the set of non-generic points. In this talk I will present a criterion for a set to have a nonempty intersection with every “regular fractal". We then apply this criterion to
show that the set of badly approximable vectors with weights intersects every regular fractal, and put it in the context of diagonalizable actions on homogeneous spaces.
This talk is based on a joint work with Dzmitry Badziahin, Stephen Harrap and David Simmons.
Bet Belgia Library, Hebrew University of Jerusalem
The Teichmüller flow acts as a renormalization operator for interval
exchange transformations. For this reason its properties give some insight
about the dynamics of rational billiards. For example Lyapunov exponents of
the Teichmüller flow are tightly related to equidistribution speed in
rational billiards. Since the mid 90's M. Kontsevich and A. Zorich started
computations of these exponents.
After giving some motivating examples for the computation of these exponents
and a brief overview of 30 years of intensive research (including works of
Manchester building, Hebrew University of Jerusalem, (Room 209)
This talk will discuss some recent developments in the study of the spectral
properties of parabolic flows and maps. More specifically, it will focus on
the techniques used to determine the spectrum of the time-changes of the
horocycle flow and and the effort to generalize these methods to create conditions
under which a general parabolic flow/map would be expected to have absolutely continuous
spectrum.
We consider self-similar Iterated Function System (IFS) on the
line constructed with overlapping cylinders. Recently there have been a
number of outstanding results which have suggested that the overlap has
dramatic change in the most important properties of the IFS only if there is
an exact overlap between some of the cylinders.
In this talk, we point out that this is not always the case, at least as far
as the absolute continuity of self-similar measures is concerned.
Namely, we present some one-parameter families of homogeneous self-
Manchester building, Hebrew University of Jerusalem, (Room 209)
Consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane.
Consider a real Gaussian stationary process, either on Z or on R. That is,
a stochastic process, invariant under translations, whose finite marginals
are centered multi-variate Gaussians. The persistence of such a process on
[0,N] is the probability that it remains positive throughout this interval.
The relation between the decay of the persistence as N tends to infinity
and the covariance function of the process has been investigated since the
1950s with motivations stemming from probability, engineering and
Manchester building, Hebrew University of Jerusalem, (Room 209)
I describe a language and set-up for proving monotonicity of entropy for families of interval maps which are defined locally. This can be seen as a local version of Thurston's algorithm. We apply this approach to prove the monotonicity and related results for families that are
not covered by other methods (with flat critical point, piecewise linear, Lorenz-type, Arnold family and others) . Joint work with Weixiao Shen and Sebastian van Strien.
It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.
Manchester building, Hebrew University of Jerusalem, (Room 209)
It is well known that for almost every x in (0,1) its orbit under the Gauss map, namely T(x)=1/x-[1/x], equidistributes with respect to the Gauss-Kuzmin measure. This claim is not true for all x, and in particular it is not true for rational numbers which have finite "orbits" which terminate in 0. In order to still have some equidistribution, we instead group
together the orbits corresponding to p/q when q is fixed and (p,q)=1 and ask whether these finite sets equidistribute as q goes to infinity.
