Decay of correlations in one-dimensional dynamics
Density of paths of iterated Lévy transforms of brownian motion
The Lévy transform of a Brownian motion B is the Brownian motion B(1) given by Bt(1) = ∫0tsgn(Bs)dBs; call B(n) the Brownian motion obtained from B by iterating n times this transformation. We establish that almost surely, the sequence of paths (t → Bt(n))n⩾0 is dense in Wiener space, for the topology of uniform convergence on compact time intervals.
Density of paths of iterated Lévy transforms of Brownian motion
The Lévy transform of a Brownian motion B is the Brownian motion B(1) given by Bt(1) = ∫0tsgn(Bs)dBs; call B(n) the Brownian motion obtained from B by iterating n times this transformation. We establish that almost surely, the sequence of paths (t → Bt(n))n⩾0 is dense in Wiener space, for the topology of uniform convergence on compact time intervals.
Density of the set of symbolic dynamics with all ergodic measures supported on periodic orbits
Let K be the Cantor set. We prove that arbitrarily close to a homeomorphism T: K → K there exists a homeomorphism T̃: K → K such that the ω-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism T: K → K there exists an endomorphism T̃: K → K with every orbit finally periodic.
Diophantine classes, dimension and Denjoy maps
Disjointness properties for Cartesian products of weakly mixing systems
For n ≥ 1 we consider the class JP(n) of dynamical systems each of whose ergodic joinings with a Cartesian product of k weakly mixing automorphisms (k ≥ n) can be represented as the independent extension of a joining of the system with only n coordinate factors. For n ≥ 2 we show that, whenever the maximal spectral type of a weakly mixing automorphism T is singular with respect to the convolution of any n continuous measures, i.e. T has the so-called convolution singularity property of order n,...
Dispersing cocycles and mixing flows under functions
Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X,,μ) and A a locally compact second countable abelian group. A cocycle c: G × X → A for T disperses if in measure for every map α: G → A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A₀ ⊂ A such that the composition θ ∘ c: G × X → A/A₀ of c with the quotient map θ: A → A/A₀ is trivial (i.e. cohomologous to a homomorphism η: G → A/A₀). This result...
Dispersion properties of ergodic translations.
Dynamics of finite-multivalued transformations.
Dynamics via measurability.
Dynamiques associees a une echelle de numeration