We give sufficient conditions for asymptotic stability of a Markov operator governing the evolution of measures due to the action of randomly chosen dynamical systems. We show that the existence of an invariant measure for the transition operator implies the existence of an invariant measure for the semigroup generated by the system.

For a dynamical system (X,T,μ), we investigate the connections between a metric invariant, the rank r(T), and a spectral invariant, the maximal multiplicity m(T). We build examples of systems for which the pair (m(T),r(T)) takes values (m,m) for any integer m ≥ 1 or (p-1, p) for any prime number p ≥ 3.

We prove ratio Tauberian theorems for relatively bounded functions and sequences in Banach spaces.

A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

We consider some descriptive properties of supports of shift invariant measures on ${ℂ}^{\mathbb{Z}}$ under the assumption that the closed linear span (in $L^2$) of the co-ordinate functions on ${ℂ}^{\mathbb{Z}}$ is all of $L^2$.

Soit $T$ un opérateur linéaire positif sur $𝒞\left(K\right)$ (où $K$ est un compact). On montre que si inf. $\{T_1^n;n\>0\}\<1$, la suite des $(T^n$) converge uniformément vers 0, et que si sup. $\{T_1^n;n\>0\}\>1$ la suite des $\left(T^n\right)$ converge uniformément vers $+\infty$.Puis on applique ces deux énoncés à l’étude des suites : $\left[{\sum }_0^{n-1}{T}^{i}f\right]/n$ et $(T^{n}f{)}^{1/n}$ ; on donne en particulier plusieurs critères de convergence uniforme de ces suites.

The notion of local mean ergodicity is introduced. Some general locally mean ergodic theorems for linear and affine operators are presented. Locally mean ergodic theorems for affine operators whose linear parts are compact or similar to subnormal operators on a Hilbert space are given.

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ\left(U\right)}_{n=1}^{\infty }$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ\left(U\right)=1/n{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))U^k$. We show that this sequence ${ₙ\left(U\right)}_{n=1}^{\infty }$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

For a bounded linear operator T in a Banach space the Ritt resolvent condition $\parallel R_{\lambda }\left(T\right)\parallel \le C/|\lambda -1|$ (|λ| > 1) can be extended (changing the constant C) to any sector |arg(λ - 1)| ≤ π - δ, $arccos\left(C^{-1}\right)<\delta <\pi /2$. This implies the power boundedness of the operator T. A key result is that the spectrum σ(T) is contained in a special convex closed domain. A generalized Ritt condition leads to a similar localization result and then to a theorem on invariant subspaces.