Strictly ergodic models and topological mixing forZ 2-action

We prove that any free ergodicZ ²-action has a strictly ergodic and topologically mixing model.     

Spectra of ergodic transformations

We study the group properties of the spectrum of a strongly continuous unitary representation of a locally compact Abelian group G implementing an ergodic group of ¹-automorphisms of a von Neumann algebra $\text{R}$. It is shown that in many cases the spectrum equals the dual group of G; e.g. if G is the integers and $\text{R}$ not finite dimensional and Abelian, then the spectrum is the circle group.     

Jointly ergodic measure-preserving transformations

The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that ifT ₁,T ₂, …,T _s are invertible commuting measure-preserving transformations of a probability space (X, ?, μ) then 1 $$\frac{1}{{N - M}}\sum\limits_{n = M}^{N - 1} {T{}_1^n } f_1 .T_2^n f_2 .....T_s^n f_s \xrightarrow[{N - M \to \propto }]{{I^2 (X)}}(\int_X {f1d\mu )} (\int_X {f2d\mu )...(\int_X {fsd\mu )} } $$ for anyf ₁,f ₂, …,f _s∈L ^x (X, ?, μ) iffT ₁×T ₂×…×T _s and all the transformationsT _iT_j ¹,i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.     

Rigid factors of ergodic transformations

We prove a theorem concerning cartesian products of ergodic not necessarily measuring preserving transformations, using the notion of rigid factors for such transformations.     

Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1 transformations

The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III₁ ergodic transformations form a denseG set with respect to the coarse topology. They also prove that for any locally compact second countable abelian groupH, and any ergodic type III transformationT, it is generic in the space ofH-valued cocycles for the integer action given byT that the skew product ofT with the cocycle is orbit equivalent toT. Similar results are given for ergodic measure-preserving transformations as well.Research supported in part by: Nat. Sci. and Eng. Res. Council #A7163 and # U0080 F.C.A.C. Quebec, NSF Grants # MCS-8102399 and # DMS-8418431.     

Approximation for the measures of ergodic transformations of the real line

Summary Let be the Frobenius-Perron operator corresponding to a nonsingular point-transformation of the real lineR into itself and let for each natural numbern, P _n be the discrete analogue ofP _f. It is shown that under fairly weak restrictions on, the equationf=P _f f has an unique solutionf ₀ such thatf ₀>0 (a.e.), ¦f ₀¦=1, and that this solution can be approximated inL ¹ (R) in two different ways: (1) by the sequence wheref0, ¦f¦=1, and (2) by the sequence {s _On } of simple functions such thats _n=P_n(s _On ).     

Minimal models for noninvertible and not uniquely ergodic systems

Let (Y, S) be a (not necessarilly invertible) topological dynamical system on a zero-dimensional metric spaceY without periodic points. Then there exists a minimal system (X, T) with the same simplex of invariant measures as (Y, S). More precisely, there exists a Borel isomorphism between full sets inY andX such that the adjoint map on measures is a homeomorphism between the corresponding sets of invariant measures in the weak topology. As an application we construct a minimal system carrying isomorphic copies of all nonatomic invariant measures.     

Adaptive efficient analysis for big data ergodic diffusion models

Statistical Inference for Stochastic Processes - We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete...     

On the simplicity of the full group of ergodic transformations

LetE denote an invertible, non-singular, ergodic transformation of (0, 1). Then the full group ofE is perfect. IfE preserves the Lebesgue measure, then the full group is simple. IfE preserves no measure equivalent to Lebesgue, then the full group is simple. IfE preserves an infinite measure, then there exists a unique normal subgroup. IfT is any invertible transformation preserving the Lebesgue measure, then the full group is simple if and only ifT is ergodic on its support.     

On the pointwise ergodic behaviour of transformations preserving infinite measures

We consider situations in which the asymptotic type of a measure preserving transformation manifests itself in a pointwise manner.