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Generic extensions of ergodic systems V. V. Ryzhikov Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 10, DOI: https://doi.org/10.4213/sm9844 
Bibliographic databases:
    § 1. IntroductionThe automorphism group $ \operatorname{Aut}=\operatorname{Aut}(\mu)$ of a standard probability space $(X,\mathcal B,\mu)$ is endowed with the complete Halmos metric $\rho $. The distance between automorphisms $S$ and $T$ is defined by
$$
\begin{equation*}
\rho(S,T)=\sum_i 2^{-i}\bigl(\mu(SA_i\Delta TA_i)+\mu(S^{-1}A_i\Delta T^{-1}A_i)\bigr),
\end{equation*}
\notag
$$
where $\{A_i\}$ is some fixed family of sets that is dense in the algebra $\mathcal B$. A set of automorphisms is said to be generic if it contains some $G_\delta$-set that is dense in $\operatorname{Aut}$. A property of automorphisms is said to be generic if the set of automorphisms with this property is generic. When experts use to say that a ‘generic automorphism’ is ergodic, they actually mean precisely that the set of ergodic automorphisms is generic. We also use this mathematical slang. The theory of generic actions with invariant measures has a long history, finds applications, for example, in the spectral theory of dynamical systems (see [1]), and continues to attract the attention of researches, which is evidenced by recent works (see [2]–[9]). It was shown in [6] that generic extensions preserve the $K$-property and the Bernoulli property of the base; it was established in [9] that an ergodic transformation $S$ with positive entropy is isomorphic to its generic extension (the word transformation is used as a synonym of automorphism). On the contrary, a transformation with zero entropy is not isomorphic to its generic extension (see [9]). We give another proof of this fact using the method from [7]: given an automorphism $S$, a numerical invariant $h_P$ (like the Kushnirenko entropy: see [10]) can be chosen so that $h_P(S)=0$ but $h_P(R)=\infty$ for generic extensions $R$ of $S$. An automorphism that is isomorphic to its inverse is called symmetric. Symmetry is preserved under actions with positive entropy, which immediately follows from the result on isomorphisms in [9]. We show that this property is not preserved for generic extensions of a rigid symmetric automorphism. Some spectral and asymptotic invariants are inherited by generic extensions: we show that these are the singularity of the spectrum of an automorphism, partial rigidity, and mild and strong mixing. The theory of generic actions can formally be viewed as a theory of generic extensions of the identity action on a one-point space. If the base measure space consists of a finite number of points, then in considering extensions of the identity action we study the generic properties of a finite set of actions. Generic extensions of the identity transformation defined on a standard Lebesgue space characterize generic measurable families of transformations having the cardinality of the continuum. In particular, we show the following. Given a generic family $\{T_x\}$, there exists a sequence $m_i\to\infty$ such that the powers $T^{m_i}_x$ converge to the identity operator (dynamic conformism). For another sequence $n_j\to\infty$ the powers $T^{n_i}_x$ converge weakly to operator-valued polynomials $P_x(T_x)$, were all operators $P_x$ are distinct (dynamic individualism). Topics in generic extensions are many, since for each invariant of a measure-preserving action of a given group there is a problem of lifting it by a generic extension arises. Some general problems are yet to be solved. Does a generic property of an automorphism descend on its nontrivial factors? Does a generic extension have an intermediate extension? Are the Lebesgue property of the spectrum and the multiple mixing property stable? Similar questions arise when spaces of actions preserving a fixed subalgebra are considered and relative invariants are investigated (see [3]). Generic extensions of actions on spaces with sigma-finite measures are of interest. A range or problems concerning generic extensions also arises in connection with complete metrics on spaces of mixing actions (see [1] and [11]). § 2. Examples of generic properties of extensionsSome asymptotic properties of an automorphism $T$ can formally be defined using a sequence of values of some function $\varphi(N,j,T)$ as $j\to\infty$. Here $N$ ranges over the set of positive integers and $\varphi(N,j,T)$ depends on $T$ continuously. We recall examples of asymptotic properties of this kind. Genericity of the weak mixing propertyIf for any sets $A,B\in\mathcal B$ we have then the automorphism $T$ is called mixing. Weak mixing means that there is a mixing sequence $j_k$, that is, the above convergence is replaced by $\mu(A\cap T^{j_k} B)\to\ \mu(A)\mu(B)$ as $k\to\infty$. We reformulate these definitions.Let the family of measurable sets $\{A_i\colon i\in\mathbb{N}\}$ be dense in $\mathcal B$; we introduce the functions
$$
\begin{equation}
\varphi(N,j,T)=\max_{1\leqslant i,k\leqslant N} |\mu(A_i\cap T^jA_k)-\mu(A_i)\mu(A_k)|.
\end{equation}
\tag{1}
$$
The mixing property is equivalent to the fact that for any $N$.The weak mixing property means that for any $N$ and $j_0$ there is $j>j_0$ such that This property is generic. We verify this. We let $U_{N,j}$ denote the set of all $T$ satisfying the last inequality. It is open. All weakly mixing automorphisms form the dense $G_\delta$-set Its density follows, for example, from the fact that the mixing automorphisms form a dense subset in $\operatorname{Aut}$ (a consequence of the classical fact that the conjugacy class of an aperiodic transformation is dense in $\operatorname{Aut}$). Halmos proved that the weak mixing property is generic, whereas Rokhlin showed that the mixing property is not generic. The latter is implied, for example, by the genericity of the rigidity property, which is incompatible with the mixing property.The genericity of the rigidity propertyWe introduce the function
$$
\begin{equation*}
\psi(N,j,T)= \max \{\mu(A_i)-\mu(A_i\cap T^jA_i)\colon 1\leqslant i \leqslant N\}.
\end{equation*}
\notag
$$
If for any $N$ and $j_0$ there is $j>j_0$ such that then the transformation $T$ is called rigid. This property is commonly stated as follows: there is a sequence $j_k\to\infty$ such that $T^{j_k}\to I$ (here weak convergence of operators coincides with strong convergence). Letting $U_{N,j}$ denote the set of all $T$ satisfying the last inequality and taking $\bigcap_{N,j_0}\bigcup_{j>j_0} U_{N,j},$ we see that the rigid transformations form a dense $G_\delta$-set. A classical example of an ergodic rigid transformation is the rotation of a circle through an angle incommensurable with $\pi$. We have shown how to establish the well-known fact that generic transformations are rigid and weakly mixing. Now we turn to the main topic of this paper. Generic extensions of automorphismsWe let $\mathbf {J}$ denote the family of all automorphisms of the space $(X\times Y, \mu\otimes\mu')$ such that sets of the form $A\times Y$ are invariant for all $A\subset X$. Automorphisms of this kind are skew products over the identity transformation $\mathrm{Id}$. We let $\operatorname{Ext}(S)$ denote all skew products $R$ over an automorphism $S$. They are also called extensions. Recall that $R$ is defined by the formula where $\{R_x\}$ is a measurable family of automorphisms of the space $(Y,\mu)$. Note that we consider only the case when $Y=X$ and $\mu'=\mu$, leaving aside finite extensions, for which $|Y|<\infty$. The Halmos metric on $\operatorname{Aut}(\mu\otimes\mu)$ induces a complete metric on the closed subspace $\operatorname{Ext}(S)$. A class of extensions containing a $G_\delta$-set that is dense in $\operatorname{Ext}(S)$ is called generic. A property of extensions is generic if all representatives of some generic class have it. To prove the genericity of a number of properties, the following well-known assertion is used, the validity of which can easily be established using the classical Rokhlin-Halmos lemma: for any skew product $R$ over an ergodic automorphism $S$ the class $\{\Phi^{-1}R\Phi \colon \Phi\in \mathbf {J}\}$ is dense in $\operatorname{Ext}(S)$. In particular, this assertion holds for direct products $R=S\times T$, where $T$ is some automorphism.Lifting the partial rigidity propertyWe fix a parameter $a\in (0,1]$ and a family $\{A_i\}$ that is dense in $\mathcal B$ to introduce the function
$$
\begin{equation*}
\psi_a(N,j,T)= \max \bigl\{a\mu(A_i)-\mu(A_i\cap T^jA_i)\colon 1\leqslant i \leqslant N\bigr\}.
\end{equation*}
\notag
$$
If for all $N$ and $j_0$ there is $j>j_0$ such that then the transformation $T$ is called $a$-rigid. This property can be formulated in another way: there is a sequence $j_k\to\infty$ such that $T^{j_k}\to_w aI+(1-a)P$, where $P$ is some Markov operator (the Markov property of an operator means that $P$ and $P^\ast$ preserve the nonnegativity of functions and map constants to themselves). Proof. An extension $R$ of $S$ has the $a$-rigidity property if for any $N$ and $j_0$ there is $j>j_0$ such that $\psi_a(N,j,R)<1/N$. Then the $G_\delta$-set consists exactly of $a$-rigid transformations. Since the product $S\times\mathrm{Id}$ and all skew products of the form $J^{-1}(S\times \mathrm{Id})J$, $ J\in\mathbf{J}$, inherit the $a$-rigidity property, $W_a$ is dense in $\operatorname{Ext}(S)$. Thus, the set $W_a$ of all $a$-rigid transformations is generic. The theorem is proved. Lifting the weak mixing propertyFor a weakly mixing automorphism $S$ we consider extensions $R$ and the function $ \varphi(N,j,R)$ defined by (1) for the measure $\mu\otimes\mu$ instead of $\mu$. If $\psi_a(N,j,R)$ is replaced by $\varphi(N,j,R)$ and the weakly mixing extension $S\times S$ is considered instead of $S\times \mathrm{Id}$ in the previous proof, then we obtain in a similar way that weakly mixing extensions form a dense $G_\delta$-set. For a skew product $R=(S,R_x)$ over an ergodic transformation $S$ the concept of relative weak mixing is defined, which means the ergodicity of the skew product $R\times_S R:=(S,R_x\times R_x)$ with respect to the measure $\mu\otimes \mu\otimes\mu$. We write $R\in \operatorname{RWM}(S)$ in this case. In the general case (when we do not assume $S$ to be ergodic) relative weak mixing can be defined as follows: for any $A,B\in\mathcal B$
$$
\begin{equation*}
\int_X \frac 1 j \sum_{n=1}^j(\mu(C(x,n, R)A\cap B)-\mu(A)\mu(B))^2\,d\mu(x)\to 0, \qquad j\to\infty,
\end{equation*}
\notag
$$
where This property can be reformulated in the following way: the operators $C(x,n, R)$ are close to $\Theta$ for large $N$ in the weak operator topology on the average with respect to $x$ and on the average with respect to $n$, $1\leqslant n\leqslant N$. (Recall that $\Theta$ denotes the operator of orthogonal projection onto the space of constants.) We present a result from [4] due to Glasner and Weiss, along with its proof. Theorem 2.2. If $R$ is a generic extension of an ergodic automorphism $S$, then $R \in \operatorname{RWM}(S)$. Proof. We assume that a family of measurable sets $\{A_i\colon i\in\mathbb{N}\}$ is dense in $\mathcal B$ and set
The condition $R\in \operatorname{RWM}(S)$ means that for any positive integer $N$ there exists $j$ such that $ \varphi(N,j,R)<1/N$. Since $\varphi$ depends on $R$ continuously, the class $\operatorname{RWM}(S)$ is a $G_\delta$-set. The class $\operatorname{RWM}(S)$ is dense in the space $\operatorname{Ext}(S)$ because any subclass $\{J^{-1}(S\times T)J\colon J\in\mathbf {J}\}$ of it, where $T$ is a weakly mixing automorphism, is dense. The theorem is proved. § 3. Weak closures and dynamical properties of generic familiesA function $P$ of an unitary operator $T$ is said to be admissible if $P(T)=c\Theta +\sum_{i=0}^\infty c_i T^i$, where $c,c_i\geqslant 0$ and $c + \sum_{i=0}^\infty c_i=1$. Admissible weak limitsAssume that $P$ is an admissible function. Is the operator $P(R)$, where $P$ is an admissible function, the weak limit of powers of a generic extension $R$ in the case when the weak convergence $S^{n_i}\to P(S)$ takes place? The answer is in the positive for all possible factors $S$ of generic transformations $T$. Theorem 3.1. If powers of some extension $\widetilde R$ of a transformation $S$ have a weak limit $P(\widetilde R)$, then the set of all extensions $R$ with this property is generic. Proof. We consider the function where $\{f_n\}$ is a family of functions that is dense in the unit ball. Set Then $W$ is a generic set since it contains the class, dense in $\operatorname{Ext}(S)$, of extensions that are cohomologous to $\widetilde R$, that is, conjugate to $\widetilde R$ via some $\Phi\in\mathbf {J}$. At the same time $W$ contains all $R$ such that the weak closure of their powers contains the operator $P$. The theorem is proved. Remark 3.1. This theorem can be improved by replacing the single function $P$ in its statement by an arbitrary family of admissible functions. This follows directly from the fact that the set of all operators of the form $P(R)$, where $P$ ranges over a fixed family of admissible functions, is separable in the weak operator topology. Generic familiesA skew product $(\mathrm{Id}, T_x)$ over the identity operator is associated with the measurable family of automorphisms $\{T_x\colon x\in X\}$. A generic set in the space $\operatorname{Ext}(\mathrm{Id})$ is associated with a generic set of such families. Theorem 3.1 implies that almost all automorphisms in a generic family are weakly mixing. Below we establish other generic properties of measurable families in terms of weak limits. In particular, we show that, under iterations, automorphisms in a generic family converge to the identity operator along some sequences, but along other sequences part of them converge to the identity operator $I$, whereas another part converge, for instance, to the operator $\Theta$. Let us present an example. For convenience assume that $X=[0,1]$. Then for a generic family $\{T_x\colon x\in [0,1]\}$ there exists a sequence $n_i$ such that for almost all $x$.Now consider a more general situation. Let $c(x)$ and $c_i(x)$ be nonnegative measurable functions satisfying $c(x) + \sum_{i=0}^\infty c_i(x)=1$. Then for a generic family $\{T_x\colon x\in [0,1]\}$ there exists a sequence $n_i$ such that
$$
\begin{equation*}
T_x^{n_i}\to_w\ c(x)\Theta +\sum_{i=0}^\infty c_i(x) T^i_x.
\end{equation*}
\notag
$$
The above is a consequence of the following assertion. Theorem 3.2. Let $X\!=\!\bigsqcup_m A_m$, where all sets $A_m$ are of positive measure. Assume that a set of admissible functions $\{P_m\colon m\in\mathbb{N}\}$ and a sequence $n_j\to\infty$ are fixed. Then, given a generic extension of the identity operator $(\mathrm{Id},T_x)$, there exists a sequence $n_{j(k)}\to\infty$ such that for each $m$ the weak convergence $T_x^{{j(k)}}\to P_m(T_x)$ takes place for almost all $x\in A_m$. Proof. It follows from [7], Theorem 2.1, that there exists a countable family of automorphisms $U_m$ such that $U^{n_j}\to P_m(U)$ as $j\to\infty$ for some sequence ${n_j\to\infty}$. Let $R=(\mathrm{Id}, R_x)$, where $R_x=U_m$ for $x\in A_m$. The class of extensions $\{J^{-1}RJ\colon J\in\mathbf {J}\}$ is dense in $\operatorname{Ext}(\mathrm{Id})$ and is contained in the set of extensions $(\mathrm{Id}, \widetilde R_x)$ such that for each $m$ we have $\widetilde R_x^{n_j}\to P_m(\widetilde R_x)$ for $x\in A_m$. Standard arguments (like the ones in the proof of Theorem 3.1) show that the set of all extensions $W$ for which such convergence takes place for some sequence $j(k)\to\infty$ is a $G_\delta$-set. The theorem is proved.
§ 4. Generic extensions preserve the singularity of the spectrumIf a measure $\sigma$ on the unit circle $\mathbf T$ in the complex plane is singular with respect to the Lebesgue measure $m$, then this is equivalent to the following property: $(\ast)$ for any $N>0$ there is a positive integer $P$ such that
$$
\begin{equation*}
\biggl|\biggl\{k\colon \sigma(I_{k,P})<\frac{1}{NP}\biggr\}\biggr|>\biggl(1-\frac 1 N\biggr)P
\end{equation*}
\notag
$$
for the partition of the circle into the arcs $I_{k,P}=[\exp(2\pi i k/ P), \exp(2\pi i(k+1)/P)]$, $k=0, 1,\dots, P-1$. We explain why $(\ast)$ implies that the measure is singular. If $\sigma$ has an absolutely continuous component $\nu$, then for some $a,b>0$ there is a set $A$ of measure $2a$ on which the Radon-Nikodym derivative of $\nu$ is greater than $2b$. For large $P$ we approximate $A$ by unions of arcs $I_{k,P}$ in the best possible way. Most of these arcs consist mainly of points in $A$; the total measure of this majority is larger than $a$. We have deduced that for sufficiently large $P$ more than $aP$ arcs have $\sigma$-measures greater than $b/P$. In $(\ast)$ this case is forbidden for $N$ such that $1/N$ is less than $a$ and $b$. Thus, we have derived from $(\ast)$ that the measure is singular. We show that the singularity of $\sigma$ implies property $(\ast)$. Let $\sigma(\mathbf T)=1$. It follows from general facts that for any $\varepsilon>0$ and $N$ there exist $P'$ and a finite union $U$ of some arcs $I_{k,P'}$ such that $m(U)<1/(3N)$ and $\sigma(U)>1- \varepsilon/2$. For all sufficiently large $P>P'$, in addition to $U$, there is a set $V$ consisting of some arcs $I_{k,P}$ ($k$ is variable, whereas $P$ is fixed) such that $m(V)>1-1/(2N)$ and $\sigma(V)<\varepsilon$. Let $\varepsilon < 1/(2N^2)$; then for sufficiently large $P$ there are at most $P/(2N)$ arcs $I_{k,P}$ in $V$ whose $\sigma$-measures are greater than $1/(NP)$ (otherwise $\varepsilon > 1/(2N^2)$). The complement to $V$ consists of at most $P/(2N)$ arcs. Therefore, the number of arcs whose $\sigma$-measures are less than $1/(NP)$ exceeds $P-P/N$ for all sufficiently large $P$. This means that property $(\ast)$ holds. We reformulate property $(\ast)$ in terms of continuous functions on the circle. We define continuous functions $\Delta_{k,P}$ on $\mathbf T$ as follows: the function $\Delta_{k,P}$ grows linearly from 0 to 1 on the arc $I_{k-1,P}$, $k\in\mathbf {Z}_P$, is identically equal to 1 on $I_{k,P}$, decreases down to 0 linearly on $I_{k+1,P}$, and is equal to 0 on the other arcs. We set
$$
\begin{equation*}
D(\sigma,N,P)=\biggl\{k\colon \int_{\mathbf T}\Delta_{k,P}\, d\sigma < \frac 1 {NP}\biggr\}.
\end{equation*}
\notag
$$
The measure $\sigma$ on the circle $\mathbf T$ is singular only if for each $N$ there exists $P$ such that Proof. We let $\sigma_{f,R}$ denote the spectral measure of the operator $R$ with cyclic vector $f$, $\|f\|=1$, that is,
$$
\begin{equation*}
\widehat{\sigma_{f,R}}(s)=\int_{\mathbf T} z^s\,d\sigma_{f,R}=(R^sf,f).
\end{equation*}
\notag
$$
The continuous function $\Delta_{k,P}$ is uniformly close to some Fejér sum $S_n$; $\!\displaystyle\int_{\mathbf T}\!\!\Delta_{k,P}\, d\sigma_{f,R}$ is close to the integral $\displaystyle\int_{\mathbf T} S_n \,d\sigma_{f,R}$, which depends continuously on only a finite number of Fourier coefficients of the measure $\sigma_{f,R}$. Since the coefficients $\widehat{\sigma_{f,R}}$ depend on $R$ continuously, the set
$$
\begin{equation*}
\biggl\{R\colon \int_{\mathbf T}\Delta_{k,P} \,d\sigma_{f,R} < \frac 1 {NP}\biggr\}
\end{equation*}
\notag
$$
is open. Then the set $\{R\colon |D(\sigma_{f,R},N,P)|= d\}$, $d\in\mathbb{N}$, is open; hence is open too. It follows from the above that the set of automorphisms $R\in \operatorname{Aut}(\mu\otimes\mu)$ with singular measures $\sigma_{f,R}$ is the $G_\delta$-set We choose an orthonormal basis $\{f_i\}$ in the space $L_2$ and set We have proved that automorphisms $\operatorname{Aut}(\mu\otimes\mu)$ with singular spectra form a $G_\delta$-set. Skew products over $S$ form the closed set $\operatorname{Ext}(S)\subset \operatorname{Aut}(\mu\otimes\mu)$, while extensions with singular spectra form the $G_\delta$-set $\operatorname{Sing}_S$ in the induced topology on $\operatorname{Ext}(S)$. It remains to note that $\operatorname{Sing}_S$ contains the dense subset $\{J^{-1}(S\times \mathrm{Id})J \colon J\in \mathbf {J}\},$ where $\mathbf {J}$ denotes the class of skew products over $\mathrm{Id}$. The theorem is proved. The singularity of the spectrum in the theorem does not forbid the presence of a discrete component. If $S$ has a continuous singular spectrum, then its generic extension inherits this property. Remarks on spectral multiplicitiesThe preservation of some asymptotic properties (invariants) for generic extensions was proved using a method in which the central role is played by an example of a skew product with the asymptotic property in question. For partial rigidity and the singularity of the spectrum an appropriate example was the product $S\times \mathrm{Id}$. For the weak mixing property the product $S\times S$ was considered. The following theorem provides examples of extensions preserving the set of spectral multiplicities of the underlying automorphism. Theorem 4.2. If $S^{n_j}\to I$ as $n_j\to\infty$ for an automorphism $S$, then for a generic automorphism $T$ the product $R=S\times T$ has the same set of spectral multiplicities as $S$. Proof. Assume that $S$ has a simple spectrum. Among generic transformations, there exists $T$ with simple spectrum such that $T^{n_{j_k}}\to T$ for some sequence $j_k\to\infty$ (this follows from Theorem 2.1 in [7]). Then the product $S\otimes T$ has a simple spectrum, since the tensor product of cyclic vectors $f$ and $g$ of the operators $S$ and $T$ is a cyclic vector of the operator $S\otimes T$. In fact, the cyclic space $C$ with cyclic vector $f\otimes g$ for the operator $S\otimes T$ contains all vectors of the form $f\otimes T^ng$. Therefore, it contains all vectors of the form $S^mf\otimes T^{m+n}g$; hence $C$ coincides with the whole of $L_2\otimes L_2$.
If $S$ is the direct sum of a set of operators $S_i$ with simple continuous spectra and $S^{n_j}\to I$ as $n_j\to\infty$, then $S\otimes T$ is the direct sum of the operators $S_i\otimes T$, which have simple spectra for the same reason as above. If $S_i$ and $S_{i'}$ have mutually singular spectral types, then the operators $S_i\otimes I$ and $S_{i'}\otimes I$ also have mutually singular spectral types; thus, they do not admit a nonzero intertwining. Then there is no nonzero intertwining between $S_i\otimes T$ and $S_{i'}\otimes T$. We show this now. Assume that for some operator $U$; then Thus, the set of spectral multiplicities of $S_i\otimes T$ is the same as that of the operator $S$. The theorem is proved.We formulate an assertion without proof: if $S^{n_j}\to aI+ (1-a)\Theta$, $a\in (0,1]$, then a generic extension of the automorphism $S$ preserves the multiplicities of the spectrum. § 5. Asymmetry and multiple weak limitsIf two transformations $R$ and $R^{-1}$ are not isomorphic in $\operatorname{Aut}$, then $R$ is said to be asymmetric. A generic extension of a symmetric transformation with positive entropy preserves symmetry (a consequence of the result in [9]). We prove that a generic extension of a rigid transformation is asymmetric. The following assertion, which is auxiliary in view of our aims, is a simple modification of the main result in [12] (now the sequence $m_j\to\infty$ plays the role of a subsequence of a prescribed sequence). Theorem 5.1. There is a transformation $T$ with the following property: any sequence tending to infinity contains a subsequence $m_j\to\infty$ such that and for any measurable set $A$. Proof. We consider $R=S\times T$, $\overline \mu=\mu\times \mu$, where $T$ satisfies the assumptions of Theorem 5.1 and $S$ is a rigid transformation such that $S^{m_j}\to I$. Then for any $\overline A$ such that $\overline \mu(\overline A)>0$. For $\overline A_0=X \times A_0$ such that $\overline \mu(\overline A_0)=1/4$ we have
We consider the family $W$ of all $V\in \operatorname{Ext}(S)$ satisfying the following condition: for any $i$ and $j_0$ there exists $j>j_0$ such that
$$
\begin{equation*}
\overline\mu(\overline A\cap V^{m_j}\overline A \cap V^{3m_j}\overline A )> \frac 1 3 \overline\mu(\overline A)\quad\text{and}\quad \overline\mu( \overline A_0\cap V^{-m_j}\overline A_0 \cap V^{-3m_j}\overline A_0 )< \frac1{15}.
\end{equation*}
\notag
$$
The family $W$ is a $G_\delta$-set. It contains all skew products that are cohomologous to the extension $R$, that is, the class $\{J^{-1}RJ\colon J\in\mathbf {J}\}$; therefore, it is everywhere dense in $\operatorname{Ext}(S)$.
It remains to note that $W$ consists of asymmetric automorphisms. In fact, let $\Phi^{-1}V^{-1} \Phi = V\in W.$ We rewrite the last inequality in the form
$$
\begin{equation*}
\frac 1 {15} > \overline\mu( \overline \Phi^{-1} \Phi A_0\cap V^{-m_j}\Phi^{-1}\Phi \overline A_0 \cap V^{-3m_j}\Phi^{-1}\Phi\overline A_0 ).
\end{equation*}
\notag
$$
For large $j$ we have
$$
\begin{equation*}
\begin{aligned} \, \frac 1 {15} &> \overline\mu( \Phi \overline A_0\cap \Phi V^{-m_j}\Phi^{-1}\Phi \overline A_0 \cap \Phi V^{-3m_j}\Phi^{-1}\Phi\overline A_0 ) \\ &= \overline\mu( \Phi \overline A_0\cap V^{m_j}\Phi \overline A_0 \cap V^{3m_j}\Phi\overline A_0 ) > \frac 1 {12}. \end{aligned}
\end{equation*}
\notag
$$
This contradiction shows that $V$ and $V^{-1}$ are not isomorphic. The theorem is proved. Thus, we have seen that whether or not it is possible to lift the symmetry property for a generic extension depends on the mixing properties of the underlying automorphism. It seems quite plausible that symmetry is not preserved for generic extensions of all partially rigid and some mixing transformations. The result in [9] implies directly that if a symmetric ergodic transformation has positive entropy, then its generic extension preserves symmetry. In connection with this one can ask whether there is a symmetric transformation with zero entropy such that its generic extensions are also symmetric. Remark 5.1. There exist transformations $T$ with an unusual nongeneric property: all Cartesian finite powers $T\times\dots\times T$ are asymmetric, while the infinite Cartesian power $T\times T\times \cdots$ is symmetric. A hint: $T=S^{-1}\times S\times S$. § 6. Generic extensions have infinite $P$-entropyInformally, a generic transformation must mix well on extremely long time intervals. For example, mixing takes place on intervals of the form for some sequence $n_j\to\infty$. Assertion (i) in Theorem 6.1 is a particular case of Theorem 2 in [13]. Part (ii) is proved similarly to part (i).Theorem 6.1. The following assertions hold for any sequence of finite sets $P_j{\subset}\, \mathbb{N}$ getting ever farther from 0: (i) for a generic transformation $T$ there is a subsequence $j(k)\to\infty$ such that $P_{j(k)}$ is a mixing sequence; (ii) for a generic extension $R$ of a mixing transformation $S$ there is a subsequence $j(k)\to\infty$ such that $P_{j(k)}$ is a mixing sequence. Assertion (i) was developed further in terms of entropy invariants in [7]. The method from that paper is used below for generic extensions. In particular, we obtain a new proof of the result in [9] that a generic extension of a system $S$ with zero entropy is not isomorphic to $S$. We find an entropy invariant equal to 0 for this $S$ but infinite for a generic extension of the transformation $S$. Kushnirenko $P$-entropyThe definition of the $P$-entropy of an automorphism $T$ in [7] is a convenient modification of the definition of Kushnirenko sequential entropy (see [10]). Given a sequence $P$ of finite sets $P_j\subset \mathbb{N}$ and an automorphism $T$ of the probability space $(\overline X,\overline\mu)$, we define the entropy $h_P(T)$ as follows. We set
$$
\begin{equation*}
h_j(T,\xi)=\frac 1 {|P_j|} H\biggl(\bigvee_{p\in P_j}T^p\xi\biggr),
\end{equation*}
\notag
$$
where $\xi=\{C_1,C_2,\dots, C_n\}$ is a measurable partition of the set $X$. Recall that the entropy of a partition is defined by
$$
\begin{equation*}
H(\xi)=-\sum_{i=1}^n \overline\mu( C_i)\ln \overline\mu( C_i).
\end{equation*}
\notag
$$
Now set
$$
\begin{equation*}
h_{P}(T,\xi)={\limsup_j} \ h_j(T,\xi)\quad\text{and} \quad h_{P}(T)=\sup_\xi h_{P}(T,\xi).
\end{equation*}
\notag
$$
Generic transformations have poor mixing properties; this is what experts are long accustomed to. However, on some sequences, they can mix better than, for example, an automorphism in a horocycle flow $O_t$. Let
$$
\begin{equation*}
P_j=\{2^i\colon n(j)\leqslant i < n(j+1)\}, \quad\text{where } \frac{n(j+1)}{n(j)}\to \infty.
\end{equation*}
\notag
$$
It follows from the results deduced in [10] that $0<h_P(T)<\infty$ for the automorphism $T=O_1$ in the horocycle flow $O_t$. As we know, $T$ is characterized by multiple mixing and has a Lebesgue spectrum. A generic automorphism $S$ of a probability space has a singular spectrum; in addition, $S$ is rigid (that is, $S^{n_i}\to I$ for some sequence ${n_i}\to \infty$). However, as shown in [7], $h_P(S)=\infty$ for a generic automorphism. Thus, on some subsequence $P_{j(k)}$, a generic automorphism $S$ mixes much better than $T$. For convenience, below we restrict ourselves to the special case when $P_j$ is a sequence of expanding arithmetic progressions. Lemma (see [7]). If $h(S)=0$, then $h_P=0$ for some sequence If $h(T)>0$, then $h_P(S\times T)=\infty$. The first assertion follows from the fact that the entropy $h(S^j)$ is zero for any $j>0$. The second assertion of the lemma is implied by the fact that $h(T^j)=jh(T)$. Proof. We fix a family of automorphisms $\{J_q\}$, $q\in \mathbf{N}$, that is dense in $\mathbf {J}$. We set $R_q=J_q^{-1}RJ_q$, where $R=S\times T$ and $T$ is a Bernoulli automorphism with generating partition $\{C_1, C_2,\dots, C_k\}$. The family $\{R_q\}$ is dense in $\operatorname{Ext}(S)$. We consider the partition $\xi$, $H(\xi)>0$, of the set $X\times Y$ such that
We show that
$$
\begin{equation}
h_{j}(R_q,\xi)=h_{j}(R,J_q\xi) =\frac 1 {L_j} H\biggl(\bigvee_{n=1}^{L(j)} R^{nj}J_q\xi\biggr) >\frac{H(\xi)}2
\end{equation}
\tag{2}
$$
for large $j$. The skew product $J_q$ (here $q$ is fixed) has the form $J_q(x,y)=(x, Q_x y).$ The measurable family of automorphisms $\{Q_x\}$, $x\in X$, as a function of $x$ can be approximated in measure $\mu$ on $X$ by a finite-valued measurable function with values in some finite set of automorphisms $\{\widetilde Q_x\colon x\in X\}$. Here we mean that automorphisms $Q_x$ are approximated by the automorphisms $\widetilde Q_x$ with respect to the Halmos metric. Partitions of the form $\{\widetilde Q_x C_1, \dots, \widetilde Q_x C_k\}$ form a finite set; we denote these partitions by $\Delta_d$, $1\leqslant d\leqslant D$. Each $\Delta_d$ can be approximated by partitions measurable with respect to for some sufficiently large positive integer $M$. Note that for $j>2M$ the partitions $T^{nj}\eta_M$, $n=1, 2, \dots$, are independent. This means the almost independence of the partitions $\{T^{nj} \Delta_{d_n}\}$ for any choice of $d_n$, $1\leqslant d_n\leqslant D$, and, as a consequence, the almost independence of the partitions $R^{nj}J_q\xi$, $n\in \mathbb{N}$, which ensures that inequality (2) holds.
Thus, for all $q$ and $N$ there exist $j=j(q,N)$ and a neighbourhood $U(q,N)$ of the automorphism $R_q$ such that $j>N$ and for all $V\in U(q, N)$. The set is a dense $G_\delta$-set. In fact, if $ V\in W$, then for each $N$ there exists $q(N)$ such that Since $j(q(N),N)>N$, we obviously obtain the inequalityBecause we can choose Bernoulli partitions $\xi_i$ so that $H(\xi_i)\to\infty$ and the intersection of the corresponding generic sets $W_i$ is generic, it follows that $h_P(V)=\infty$ for all $V\in\bigcap_i W_i$. The theorem is proved. Exquisite familiesIn contrast to the dominance property discovered in [9] for systems with positive entropy, a family $F$ of automorphisms is said to be exquisite if for each $S\in F$ a generic set of its extensions has no representative isomorphic to some $S'\in F$. It follows from Theorem 6.2 that the set $\{S\colon h_P(S)<\infty\}$ is exquisite. § 7. Recurrence of generic cocycles, absence of an independent factor and mixing stabilityA property of an action is said to be stable if a generic extension of a system with this property also has it. As we know, the continuity of the spectrum, the singularity of the spectrum, the partial rigidity property, the deterministic nature of the system and the К-property are stable. Now we add the mild and strong mixing properties to this list. Recall that it remains an open question, whether or not the property of an automorphism to have a Lebesgue spectrum and the multiple mixing property are stable. Auxiliary assertions, which are of independent interest, are used to prove this theorem. Nonmixing factor independent of any mixing factorWe need Parreau’s theorem deduced in 2002. Theorem A (see [14]). A nonmixing ergodic automorphism has a nontrivial factor that is disjoint from all mixing automorphisms. The disjointness implies the independence of the Parreau factor from any mixing factor of the automorphism $R$; we use this below. We proceed to the proof of Theorem 7.1. Assume that an extension $R$ of a mixing automorphism $S$ is not mixing; then it has a Parreau factor, which is independent of the underlying $S$-factor. Recall that a factor is the restriction of the action to an invariant sigma-subalgebra. The stability of the mixing property is implied by the aforesaid and the following assertion. Theorem 7.2. A generic extension $R$ of a weakly mixing action $S$ does not have a nontrivial factor that is independent of the underlying $S$-factor. For a sequence of finite sets $P_j\subset\mathbb{N}$, an extension $R$ and a set $A\subset X$, $\mu(A)>0$, we set
$$
\begin{equation*}
\varphi_A(N,j,R)=\prod_{p\in P_j}\mu\biggl(x\in A\cap S^pA\colon \rho(C(x,p,R), \mathrm{Id})<\frac 1 N\biggr),
\end{equation*}
\notag
$$
where $\rho$ is the Halmos metric in the group $\operatorname{Aut}(\mu)$, and we let $C(x,n,R)$ denote the cocycle $R_{S^{n-1}x}\dots R_{Sx}R_x$. We write $R\in \mathrm{RC}(P_j,A)$ if for all $N$ and $j_0$ there exists $j>j_0$ such that $\varphi_A(N,j,R)>0$. The following assertion can be called the theorem on the recurrence of the cocycle $C(x,n,R)$ corresponding to a generic extension $R\in \operatorname{Ext}(S)$. Theorem 7.3. For any set $A$ of positive measure, a generic extension $R$ of a fixed weakly mixing transformation $S$ is in the class $\mathrm{RC}(P_j,A)$ for some sequence $P_j\subset \{j,j+1,\dots,2j\}$ such that $|P_j|/j\to 1$ as $j\to\infty$. Corollary. Given a weakly mixing transformation $S$, a generic extension $R=(S,R_x)$ of $S$ and a set $A$ of positive measure, for almost all $x\in A$ there exists a sequence $p_i\to\infty$ (which depends on $x$) such that $C(x,p_i,R)\to \mathrm{Id}$ and ${S^{p_i}(x)\in A}$. Proof of Theorem 7.3. We choose a sequence of sets $P_j$ so that $S^{p(j)}\to\Theta$ for any sequence $p(j)\to\infty$ such that $p(j)\in P_j$. Since the weak mixing property is equivalent to mixing on a set of density 1, we have additionally ensured that $|P_j|/j\to 1$ as $j\to\infty$. The class $\mathrm{RC}(P_j,A)$ is a $G_\delta$-set, which follows from its definition and the continuous dependence of $\varphi_A(N,j,R)$ on $R$. We prove that it is dense. To do this we consider the class of extensions cohomologous to the trivial extension $S\times \mathrm{Id}$. Note that For any $\varepsilon>0$ there exists a set $A'\subset A$ such that $\mu(A')>0$ and for some $x_0\in A'$ and all $x\in A'$. Then, given $N$, by choosing a sufficiently small number $\varepsilon$ we ensure that for all $x\in A'$, provided that $S^p(x)\in A'$. However, by the mixing properties of the powers $S^p$, for all sufficiently large $p\in P_j$ the set of $x\in A'$ such that $S^p(x)\in A'$ has a positive measure. The theorem is proved. Proof of Theorem 7.2. Let $E \subset X\times Y$ be independent of the $S$-factor, which means that the function $h(x)=\mu(y\colon (x,y)\in E)$ is almost everywhere equal to the measure $e$ of the set $E$. If $E$ is in an $R$-invariant algebra independent of the $S$-factor, then $h_p(x)=\mu(y\colon (x,y)\in E\cap R^pE)$ is also a constant $h_p$ almost everywhere and $h_p\to e^2$ for $p\in P_j$ as $p\to\infty$ by virtue of the mixing property of $R^p$. It follows from (3) that $h_p(x)$ is close to $e$ on a set of points $x$ of positive measure. However, $h_p(x)$ is a constant, and these constants converge to both $e$ and $e^2$. We have obtained that $e=e^2$; therefore, $\mu\otimes\mu\,(E)\in\{0,1\}$. Thus, the independent factor is trivial. Theorems 7.2 and 7.1 are proved. Theorem 7.2 also shows that the property of a system not to have rigid factors (mild mixing) is a stable property. Proof. Assume that an extension $R$ of a mildly mixing automorphism $S$ has a rigid factor that is isomorphic to an automorphism $T$ such that $T^{n_i}\to I$ as $n_i\to\infty$. Let $P$ be a Markov operator intertwining $T$ and $S$. Then we have We obtain as $n_i\to\infty$. Since $S$ has no rigid factors, the function $Pf$ must be a constant. (Otherwise, the algebra of sets generated by the function $Pf$ and its shifts $S^mPf$ is a nontrivial rigid factor.) Hence the intertwining $P$ is trivial. This means that the rigid factor is independent of the mildly mixing factor. The mild mixing property implies weak mixing. To complete the proof it remains to use Theorem 7.2. Theorem 7.4 is proved. Multiple mixing and the absence of nontrivial joinings with pairwise independenceAn automorphism $S$ mixes with multiplicity $n$ if for any $A_0, A_1,\dots,A_n\in\mathcal B$ we have
$$
\begin{equation*}
\mu\bigl(A_0\cap S^{k_1}A_1\cap S^{k_1+k_2}A_2\cap \dots\cap S^{k_1+\dots+k_n}A_n\bigr)\to \mu(A_0 )\mu(A_1 )\dotsb \mu(A_n)
\end{equation*}
\notag
$$
as $k_1,\dots, k_n\to\infty$. A self-joining of order $n>2$ with pairwise independence is an $S\times \dots\times S$-invariant measure on the cube $X^n=X\times \dots\times X$ ($n$ factors) with projections $\mu\otimes\mu$ onto all two-dimensional Cartesian faces of the cube $X^n$. If self-joinings with pairwise independence for an automorphism $S$ are trivial (coincide with $\mu^n=\mu\otimes\dots\otimes\mu$) for all $n>2$, then $S$ is said to have the $\mathrm{JR}$-property. As is known, a mixing automorphism with the $\mathrm{JR}$-property is mixing of all multiplicities. In view of the stability of the mixing property, which we have established, and the stability of the $\mathrm{JR}$-property, which was proved in [15], we arrive at the following result. Theorem 7.5. Generic extensions inherit the combination of the multiple mixing property and $\mathrm{JR}$-property of an automorphism. AcknowledgementsThe author is grateful to the referee for their comments and to E. Glasner, B. Weiss and J.-P. Thouvenot for useful discussions.
Citation in format AMSBIB
\Bibitem{Ryz23} |
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