François Ledrappier

Some properties of absolutely continuous invariant measures on an interval

We are interested in ergodic properties of absolutely continuous invariant measures of positive entropy for a map of an interval. We prove a Bernoulli property and a characterization by some variational principle.

Harmonic measures and bowen-margulis measures

We compare two families of measures defined on the absolute of the universal cover of a compact negatively-curved manifold: the harmonic measures and the Bowen-Margulis measures.

Dimension formula for random transformations

We consider compositions of random diffeomorphisms and show that the dimension of sample measures equals Lyapunov dimension as conjectured in the nonrandom case by Yorke et al.

A DIMENSION FORMULA FOR BERNOULLI CONVOLUTIONS

We present a “dynamical” approach to the study of the singularity of infinitely convolved Bernoulli measuresvβ, for β the golden section. We introducevβ as the transverse measure of the maximum entropy measure μ on the repelling set invariant for contracting maps of the square, the “fat bakers” transformation. Our approach strongly relies on the Markov structure of the underlying dynamical system. Indeed, if β=golden mean, the fat bakers transformation has a very simple Markov coding. The “ambiguity” (of order two) of this coding, which appears when projecting on the line, due to passages for the central, overlapping zone, can be expressed by means of products of matrices (of order two). This product has a Markov distribution inherited by the Markov structure of the map. The dimension of the projected measure is therefore associated to the growth of this product; our dimension formula appears in a natural way as a version of the Furstenberg-Guivarch formula. Our technique provides an explicit dimension formula and, most important, provides a formalism well suited for the multifractal analysis of this measure, as we will show in a forthcoming paper.

POISSON BOUNDARIES OF DISCRETE GROUPS OF MATRICES

If μ is a probability measure on a countable group there is defined a notion of the Poisson boundary for μ which enables one to represent all bounded μ-harmonic functions on the group. It is shown that for discrete groups of matrices this boundary can be identified with the boundary of the corresponding Lie group.

Applications of Dynamics to Compact Manifolds of Negative Curvature

There is a rick mathematical structure attached to the cobordism invariants of manifolds. In the cases described by the index theorem, a generalized cohomology theory is used to express the global properties of locally defined analytic objects. Hirzebruch’s theory of multiplicative sequences gives an expression for these invariants in terms of characteristic classes, and brings to focus their remarkable arithmetic properties. Quillen’s theory of formal groups and complex oriented cohomology theories illuminates the generalized cohomology theories themselves.

On the multifractal analysis of Bernoulli convolutions. I. Large-deviation results

We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures νγ, for γ the golden mean. In this first part we establish some large-deviation results for random products of matrices, using perturbation theory of quasicompact operators.

Sharp Estimates for the Entropy

In this paper, we consider three different exponential rates of growth associated to a symmetric random walk on a countable group: the spectral gap, the entropy and the decay of the fundamental state along the paths of the random walk. We prove general inequalities between these numbers. We hope that these inequalities, and the characterization of the cases of equality, would enable us to express fine properties of the group through rather coarse invariants.

Singularity of Projections of 2-Dimensional Measures Invariant Under the Geodesic Flow

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.

Vanishing transverse entropy in smooth ergodic theory

For a measure-preserving transformation, the entropy being zero means that there is no increasing σ -algebra. In this note, we prove that a similar phenomenon occurs for C 2 diffeomorphisms when considering the increment between the partial entropies associated with different exponents.