##
**A general condition for lifting theorems.**
*(English)*
Zbl 0748.28008

This paper gives general conditions under which ergodic properties lift from factors to extensions. Many such conditions are known for compact group extensions of weakly mixing transformations [e.g. D. J. Rudolph, Ergodic Theory Dyn. Syst. 5, 445-447 (1985; Zbl 0594.28015), or the author, Proc. Am. Math. Soc. 102, No. 1, 61-67 (1988; Zbl 0636.28007)]. In this paper the key concept is that of stable extension. First the notion of relative unique ergodicity (RUE) of an extension is defined. Then an extension \(T\) of \(S\) is said to be a stable extension if \(T\) is ergodic and \(T\times R\) is an RUE extension of \(S\times R\) for all \(R\) such that \(T\times R\) is ergodic. Equivalently, stability is defined in terms of joinings using the notion of weak stability (which is an isomorphism invariant for extensions).

Examples include ergodic compact group extensions, isometric extensions and certain affine extensions. Lifting theorems of the following type are given: Suppose \(S\) satisfies a mixing property \({\mathcal P}\) and \(T\) is an extension of \(S\) with some stability. If \(T\) is weakly mixing then it also satisfies property \(\mathcal P\). Properties that can be lifted include partial mixing, strong mixing and \(r\)-fold partial mixing.

Isometric, distal and affine extensions are studied. An \(\alpha\)-affine extension of \(S\) is a skew product of the form \(T(y,g)=(Sy,\varphi(y)\alpha(g))\), where \(\alpha:G\to G\) is a continuous group automorphism and \(\varphi:Y\to G\) is measurable. The following are shown to be equivalent for \(T\) ergodic:

(i) \(h(\alpha)=0\) (entropy of \(\alpha)\). (ii) \(T\) is a distal extension of \(S\). (iii) \(T\) is an RUE and stable extension of \(S\).

Generally stable extensions have relative entropy zero, and it is shown that in the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.

Examples include ergodic compact group extensions, isometric extensions and certain affine extensions. Lifting theorems of the following type are given: Suppose \(S\) satisfies a mixing property \({\mathcal P}\) and \(T\) is an extension of \(S\) with some stability. If \(T\) is weakly mixing then it also satisfies property \(\mathcal P\). Properties that can be lifted include partial mixing, strong mixing and \(r\)-fold partial mixing.

Isometric, distal and affine extensions are studied. An \(\alpha\)-affine extension of \(S\) is a skew product of the form \(T(y,g)=(Sy,\varphi(y)\alpha(g))\), where \(\alpha:G\to G\) is a continuous group automorphism and \(\varphi:Y\to G\) is measurable. The following are shown to be equivalent for \(T\) ergodic:

(i) \(h(\alpha)=0\) (entropy of \(\alpha)\). (ii) \(T\) is a distal extension of \(S\). (iii) \(T\) is an RUE and stable extension of \(S\).

Generally stable extensions have relative entropy zero, and it is shown that in the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.

Reviewer: G.R.Goodson (Towson)

### MSC:

28D05 | Measure-preserving transformations |

54H20 | Topological dynamics (MSC2010) |

28D20 | Entropy and other invariants |

### Keywords:

stable extension; relative unique ergodicity; ergodic compact group extensions; isometric extensions; affine extensions; lifting theorems; weakly mixing; strong mixing; entropy; continuous flow extensions; ergodic homeomorphisms
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\textit{E. A. Robinson jun.}, Trans. Am. Math. Soc. 330, No. 2, 725--755 (1992; Zbl 0748.28008)

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