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A probabilistic version of the Frequent Hypercyclicity Criterion

Sophie Grivaux — 2006

Studia Mathematica

For a bounded operator T on a separable infinite-dimensional Banach space X, we give a "random" criterion not involving ergodic theory which implies that T is frequently hypercyclic: there exists a vector x such that for every non-empty open subset U of X, the set of integers n such that Tⁿx belongs to U, has positive lower density. This gives a connection between two different methods for obtaining the frequent hypercyclicity of operators.

IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products

Sophie Grivaux — 2013

Studia Mathematica

If ( n k ) k 1 is a strictly increasing sequence of integers, a continuous probability measure σ on the unit circle is said to be IP-Dirichlet with respect to ( n k ) k 1 if σ ̂ ( k F n k ) 1 as F runs over all non-empty finite subsets F of ℕ and the minimum of F tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have recently been investigated by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz products.

Invariant subspaces of X * * under the action of biconjugates

Sophie GrivauxJan Rychtář — 2006

Czechoslovak Mathematical Journal

We study conditions on an infinite dimensional separable Banach space X implying that X is the only non-trivial invariant subspace of X * * under the action of the algebra 𝔸 ( X ) of biconjugates of bounded operators on X : 𝔸 ( X ) = { T * * T ( X ) } . Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of c 0 , and show in particular that any space which does not contain 1 and has property (u) of Pelczynski is simple.

Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

Sophie GrivauxMaria Roginskaya — 2013

Czechoslovak Mathematical Journal

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle 𝕋 . A set of integers is called r -Bohr if it is recurrent for all products of r rotations on 𝕋 , and Bohr if it is recurrent for all products of rotations on 𝕋 . It is a result due to Katznelson that for each r 1 there exist sets of integers which are r -Bohr but not ( r + 1 ) -Bohr. We present new examples of r -Bohr sets which are not Bohr, thanks to a construction which...

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