Sequences and dynamical systems associated with canonical approximation by rationals
Sets of -expansions and the Hausdorff measure of slices through fractals
We study natural measures on sets of -expansions and on slices through self similar sets. In the setting of -expansions, these allow us to better understand the measure of maximal entropy for the random -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...
Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle . A set of integers is called -Bohr if it is recurrent for all products of rotations on , and Bohr if it is recurrent for all products of rotations on . It is a result due to Katznelson that for each there exist sets of integers which are -Bohr but not -Bohr. We present new examples of -Bohr sets which are not Bohr, thanks to a construction which...
Stalking staggeringly large numbers
Strong convergence of additive Multidimensional Continued Fraction algorithms
Symbolic discrepancy and self-similar dynamics
We consider subshifts arising from primitive substitutions, which are known to be uniquely ergodic dynamical systems. In order to precise this point, we introduce a symbolic notion of discrepancy. We show how the distribution of such a subshift is in part ruled by the spectrum of the incidence matrices associated with the underlying substitution. We also give some applications of these results in connection with the spectral study of substitutive dynamical systems.
Symmetric cocycles and classical exponential sums