Abstract. We classify measures on the locally homogeneous space Γ \ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in the proof of the main result. 1.

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally C∞-rigid (up to an automorphism). This result is the main part in the proof of local C∞-rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil–manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper “non–stationary” generalization of the classical theory of normal forms for local contractions.

We associate via duality a dynamical system to each pair (R S , #), where R S is the ring of S--integers in an A--field k, and # is an element of R S \{0}.

The first part of the paper begins with an introduction into Anosov actions of Z k and R k and an overview of the method of studying invariant measures for such actions based on consideration of conditional measures along various invariant foliations. The main body of that part contains a detailed proof of a modified version of the main theorem from [KS3] for actions by toral automorphisms of with applications to rigidity of the measurable structure of such actions with respect to Lebesque measure. In the second part principal technical tools for studying nonuniformly hyperbolic actions of Z k and R k are introduced and developed. These include Lyapunov characteristic exponents, nonstationary normal forms and Lyapunov Hoelder structures. At the end new rigidity results for Z² actions on three-dimensional manifolds are outlined.

In financial market risk measurement, Value-at-Risk (VaR) techniques have proven to be a very useful and popular tool. Unfortunately, most VaR estimation models suffer from major drawbacks: the lognormal (Gaussian) modeling of the returns does not take into account the observed fat tail distribution and the non-stationarity of the financial instruments severely limits the efficiency of the VaR predictions. In this paper, we present a new approach to VaR estimation which is based on ideas from the field of information theory and lossless data compression. More specifically, the technique of context modeling is applied to estimate the VaR by conditioning the probability density function on the present context. Tree-structured vector quantization is applied to partition the multi-dimensional state space of both macroeconomic and microeconomic priors into an increasing but limited number of context classes. Each class can be interpreted as a state of aggregation with its own statistical and dynamic behavior, or as a random walk with its own drift and step size. Results on the US S&P500 index, obtained using several evaluation methods, show the strong potential of this approach and prove that it can be applied successfully for, amongst other useful applications, VaR and volatility prediction. The October 1997 crash is indicated in time.

In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lower-dimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective low-dimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVD-based methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptotics-based mode elimination, coarse timestepping methods and transfer-operator based methodologies.

A non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they are coded by an arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

We propose a novel approach for building nite memory predictive models similar in spirit to variable memory length Markov models (VLMMs). The models are constructed by rst transforming the n-block structure of the training sequence into a geometric structure of points in a unit hypercube, such that the longer is the common sux shared by any two n-blocks, the closer lie their point representations.

Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.

We have recently shown that when initialized with "small" weights, recurrent neural networks (RNNs) with standard sigmoid-type activation functions are inherently biased towards Markov models, i.e. even prior to any training, RNN dynamics can be readily used to extract finite memory machines (Hammer & Tino, 2002; Tino, Cernansky & Benuskova, 2002; Tino, Cernansky & Benuskova, 2002a). Following Christiansen and Chater (1999), we refer to this phenomenon as the architectural bias of RNNs. In this paper we further extend our work on the architectural bias in RNNs by performing a rigorous fractal analysis of recurrent activation patterns. We assume the network is driven by sequences obtained by traversing an underlying finite-state transition diagram -- a scenario that has been frequently considered in the past e.g. when studying RNN-based learning and implementation of regular grammars and finite-state transducers. We obtain lower and upper bounds on various types of fractal dimensions, such as box-counting and Hausdor# dimensions. It turns out that not only can the recurrent activations inside RNNs with small initial weights be explored to build Markovian predictive models, but also the activations form fractal clusters the dimension of which can be bounded by the scaled entropy of the underlying driving source. The scaling factors are fixed and are given by the RNN parameters.