Hillel Furstenberg

Random matrix products and measures on projective spaces

The asymptotic behavior of ‖XnXn−1…X1υ‖ is studied for independent matrix-valued random variablesXn. The main tool is the use of auxiliary measures in projective space and the study of markov processes on projective space.

Ergodic fractal measures and dimension conservation

A linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For ‘homogeneous’ fractals (to be defined), there is a phenomenon of ‘dimension conservation’. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This ‘almost everywhere’ result implies a non-probabilistic statement for homogeneous fractals.

Ergodic Theory and Configurations in Sets of Positive Density

We shall present here two examples from “geometric Ramsey theory” which illustrate how ergodic theoretic techniques can be used to prove that subsets of Euclidean space of positive density necessarily contain certain configurations. Specifically we will deal with subsets of the plane, and our results will be valid for subsets of “positive upper density”.

IP-sets and polynomial recurrence

We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincare sets for measure-preserving systems, that is, sets of integers along which the analogue of the Poincare recurrence theorem holds. This is done by applying to measure-preserving transformations a limit theorem for products of appropriate powers of a commuting family of unitary operators.

On almost 1-1 extensions

We show that a broad class of extensions of measure preserving systems in the context of ergodic theory can be realized by topological models for which the extension is “almost one-one”.

Piecewise-Bohr Sets of Integers and Combinatorial Number Theory

We use ergodic-theoretical tools to study various notions of “large” sets of integers which naturally arise in theory of almost periodic functions, combinatorial number theory, and dynamics. Call a subset of N a Bohr set if it corresponds to an open subset in the Bohr compactification, and a piecewise Bohr set (PWB) if it contains arbitrarily large intervals of a fixed Bohr set. For example, we link the notion of PWB-sets to sets of the form A+B, where A and B are sets of integers having positive upper Banach density and obtain the following sharpening of a recent result of Renling Jin.

Markov Processes and Ramsey Theory for Trees

We consider analogues of van der Waerdens theorem and Szemeredis theorem, where arithmetic progressions are replaced by binary trees with a fixed distance between successive vertices. The proofs are based on some novel recurrence properties for Markov processes.

Ergodic theory and fractal geometry

Introduction to fractals Dimension Trees and fractals Invariant sets Probability trees Galleries Probability trees revisited Elements of ergodic theory Galleries of trees General remarks on Markov systems Markov operator

Ergodic Structures and Non-conventional Ergodic Theorems

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Recurrence for Stationary Group Actions

and measure preserving transformation