Sergiy Borodachov ,
On a certain separation condition for attractors of finite systems of contractive homeomorphisms.

We consider a certain separation condition for attractors of
finite systems of contractive homeomorphisms of a complete metric space,
which is in general weaker than the strong open set condition and is not
equivalent to the weak separation property.
We show that the set of
N-tuples of contractive homeomorphisms,
which satisfy this separation condition is a G_delta
set in the topology of pointwise convergence of every
component mapping with the requirement that contraction coefficients
of all mappings in the sequence be bounded
from above by a number less than one.
We give several sufficient conditions for this separation property.
Certain density results for sets of parameters defining N-tuples
of contracive similitudes on the real line,
which satisfy this separation condition, are also obtained.
This is a joint work with
Tim Bedford and
Jeff Geronimo .

Nikolai Chernov and
Dmitry Dolgopyat ,
Anomalous current in periodic Lorentz gases with infinite horizon.

We study electrical current in two-dimensional periodic Lorentz gas
with infinite horizon in the presence of a weak electric field. We
find that infinite corridors between scatterers cause a failure of
classical Ohm's law, i.e. the current J is no longer
proportional to the voltage difference E instead, it is given
by

J= (D |E log ||E|||)/2 +O(E)

where D is the `superdiffusion' matrix of the
Lorentz particle moving freely without electrical fields. The last
fact means that the Einstein relation (suitably interpreted) holds.
We also obtain new results for the infinite horizon Lorentz gas
without external fields, complementing recent studies by Szasz and
Varju.

We consider the discrete dynamical system (A, T), where A is a class of smooth real functions defined on some interval and T: A-->A is an operator Th=f(h) where is a function on the real line. Iteration of smooth maps appears naturally in the study of continuous difference equations and boundary value problems such as those in Romanenko et al [2], Severino et al [3], Sharkovsky et al [4]-[5], Vinagre et al [6]. In our work we develop some techniques of symbolic dynamics for the discrete dynamical system (A,T). We analyze in detail the case in which T arises from unimodal interval maps. This is a joint work with Carlos C. Ramos and Sandra Vinagre

- Correia, M. F., Ramos, C. C. and Vinagre, S., On the iteration of smooth maps, Accepted to be published in World Scientific Publishing, (2008).
- Romanenko, E. Yu. and Sharkovsky, A. N., From boundary value problems to difference equations: a method of investigation of chaotic vibrations, Internat J. Bifur. Chaos Appl. Sci. Engrg. 9, 7, (1999), 1285-1306.
- Severino, R., Sharkovsky, A., Sousa Ramos, J. and Vinagre, S., Symbolic Dynamics in Boundary Value problems, Grazer Math. Ber, 346, (2004), 393-402
- Sharkovsky, A. N.; Maistrenko, Yu. L.; Romanenko, E. Yu. Difference equations and their applications. Translated from the 1986 Russian original by D. V. Malyshev, P. V. Malyshev and Y. M. Pestryakov. Mathematics and its Applications, 250. Kluwer Academic Publishers Group, Dordrecht, 1993. xii+358 pp.
- Sharkovsky, A. Difference equations and boundary value problems. Proceedings of the Sixth International Conference on Difference Equations, 3-22, CRC, Boca Raton, FL, 2004.
- Vinagre, Sandra; Severino, Ricardo; Ramos, J. Sousa invariants in nonlinear boundary value problems. Chaos Solitons Fractals 25 (2005), no. 1, 65-78.

We present scaling laws for the average bursting time for a type of bifurcation of an attractor in an invariant manifold, assuming the dynamics within the manifold to be hyperbolic. This type of global bifurcation appears in nearly synchronized deterministic and random systems, and is conjectured to be typical among those breaking the asymptotic stability of a hyperbolic invariant manifold. Andrew Tö rö k , Extreme value theory for non-uniformly expanding dynamical systems systems

For an observation A:X-->R over a (discrete or continuous) measure preserving dynamical system f_t: X--> X, consider the maxima along trajectories

M_t(x):=max_{[0,t]} A(f_s x)) .

One is interested in the existence of constants a_t, b_t such that a_t(M_t-b_t) has a nondegenerate limit in distribution. The iid case is well understood: there are only three types of possible limits, and the limit is determined by the distribution of A. P. Collet (2001) proved that for a discrete system modeled by a Young Tower with exponential tail and A(x):=-log dist(x,x_0), the extreme value distribution exists for almost each x_0, and the limit is the same as if the observations were iid. We establish extreme value statistics for functions with multiple maxima and some degree of regularity on certain non-uniformly expanding dynamical systems. We also establish extreme value statistics for time-series of observations on discrete and continuous suspensions of the discrete dynamical system. We also establish extreme value statistics for time-series of observations on discrete and continuous suspensions of certain non-uniformly expanding dynamical systems via a general lifting theorem. The main result is that a broad class of observations on these systems exhibit the same extreme value statistics as iid processes with the same distribution function. This is a joint work with Mark Holland and Matt Nicol