Nonsingular transformations that are ergodic with isometric coefficients and not weakly doubly ergodic

The notion of weak mixing for finite measure-preserving transformations has several different and equivalent characterizations, and each has played different roles in various applications of the weak mixing property. It is now well-known that in infinite measure the theory is quite different. The first example showing this was given by Kakutani and Parry when they constructed in [15] infinite measure-preserving Markov shifts $T$ such that $T\times T$ is ergodic but $T\times T\times T$ is not ergodic (as is well-known, this cannot happen for finite measure-preserving transformations). Since then many related notions an examples have been constructed for infinite measure-preserving and nonsingular transformations; we refer to [5] for a survey of these results and to [13] where many of these results are discussed in the context of group actions.

While our first examples are infinite measure-preserving transformations we also consider a nonsingular versions of our construction.

Acknowledgments: The research for this paper started during the 2018 SMALL undergraduate research project at Williams College, where the first-named author was part of the ergodic theory group. We thank the other members of the ergodic theory group, which included Hindy Drillick, Alonso Espinosa-Dominguez, Jennifer N. Jones-Baro, and Yelena Mandelshtam. Support for the project was provided by National Science Foundation grant DMS-1659037 and the Science Center of Williams College. The first-named author was also supported by Williams College in summer 2019. We would like to thank Terrence Adams, and an anonymous referee for comments and suggestions on our work. Part of this work, in particular Section 5 is based on the undergraduate thesis at Williams College of BH, supervised by the third-named author. From August 2019 until August 2021, CS has been serving as a Program Director in the Division of Mathematical Sciences at the National Science Foundation (NSF), USA, and as a component of this job, he received support from NSF for research, which included work on this paper. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

We let $(X,\mathcal{B},\mu)$ denote a standard Borel space that we will assume to be nonatomic, where $\mu$ is a $\sigma$-finite measure, often assume to be a probability measure; sometimes we may omit the $\sigma$-algebra $\mathcal{B}$ from the notation. If we start with a $\sigma$-finite infinite measure $(X,\mu)$ we can always choose an equivalent probability measure $\nu$. In all cases we assume the measures are nonatomic. A transformation $(X,\nu,T)$ (sometimes we may simply write the transformation $T$) is nonsingular if it is measurable ($T^{-1}(A)\in\mathcal{B}$ for all $A\in\mathcal{B})$ and $\mu(A)=0$ if and only if $\mu(T^{-1}(A))=0$. A transformation $T$ is ergodic if whenever $\mu(T^{-1}A\bigtriangleup A)=0$ we have $\mu(A)=0$ or $\mu(A^{c})=0$, and it is conservative if for all sets $A$ of positive measure there exists an integer $n>0$ with $\mu(A\cap T^{-n}A)>0$. We will assume our transformations are invertible (i.e., invertible on a set of full measure and with measurable inverse); in this case, as the measures are nonatomic, it is known that if a transformation is ergodic, then it is conservative, see e.g. [19, 3.9.1].

A nonsingular transformation $(X,\mu,T)$ is doubly ergodic (DE) if $T\times T$ is ergodic. It is weakly doubly ergodic (WDE) if for any two measurable sets $A$, $B$ with positive measure, there exists $n\in\mathbb{Z}$ such that $\mu(T^{n}A\cap A)>0$ and $\mu(T^{n}A\cap B)>0$ (we note that this notion was originally called doubly ergodic in [8], and that we can choose $n>0$ in the definition). The transformation $T$ is ergodic with isometric coefficients (EIC—see [13]) if for any separable metric space $Y$ and $S:Y\to Y$ an invertible isometry, for any equivariant Borel map $\phi:(X,\nu,T)\to Y$, i.e., $S\circ\phi=\phi\circ T$ $\mu$-a.e., the image of almost all points in $X$ under $\phi$ is a single point., i.e., the map $\phi$ is constant a.e. A transformation $T$ is weakly mixing (WM) if for any ergodic measure preserving transformation on a probability space $(Y,m,S)$, the transformation $T\times S$ is ergodic on $X\times Y$ with respect to the product measure $\mu\otimes m$. All these notions only depend on the measure class of the measure. We use the following result from in [3].

From the definitions it follows that if a transformation is DE then it is WDE; the fact that the converse is not true for infinite measure-preserving transformations was shown in [4, 8]. Glasner and Weiss [12] proved that DE implies EIC and that EIC implies WM, and later Loh and Silva proved that WDE implies EIC (these results are in the context of nonsingular group actions). In [12], it is shown that for nonsingular group actions, WM implies ergodic; for the case of nonsingular transformations $T$ WM implies WM of $T^{n},n\neq 0$ [16, 2.1]. For completeness we observe that EIC implies ergodicity: if a nonsingular transformation $T$ is EIC and $A$ is a $T$-invariant set we can define a map $\phi:X\to\{0,1\}$ by $\phi(x)=0$ for $x\in A$ and $\phi(x)=1$ for $x\in A^{c}$. Let $S$ be the identity on $\{0,1\}$. Then $\phi$ is a non-constant a.e. equivariant map onto the isometry $S$ on $\{0,1\}$; total ergodicity of $T$ can also be shown by a similar argument.

Glasner and Weiss [12] also showed that in infinite measure EIC does not imply DE. In this paper we prove that in infinite measure EIC does not imply WDE. Glasner and Weiss [13] also asked whether WM implies EIC, which remains open as far as we know; in Section 5 we study some properties related to this question. In summary, we now know the following implications and that the converse does not hold for each of the first two implications.

We have already mentioned that WDE implies EIC; in this section we show that the converse of this implication is not true; our approach is to develop some techniques to verify the EIC property for some transformations.

The example is a rank-one transformation, first defined by Adams, Friedman and Silva in [4], that has been called the HK(+1) transformation since it is a variant of the Hajian-Kakutani transformation. It is a rank-one tower transformation. For the base case choose column $C_{0}$ consisting of the unit interval $[0,1)$, and let $r_{k}=2,s(k,0)=0$ and $s(k,1)=2h_{k}+1$, for all $k\geq 0$. So the height is given by $h_{k+1}=4h_{k}+1$. This defines an infinite measure-preserving rank-one transformation. In [4], it was shown to be weakly mixing but not doubly ergodic; the proof in [4] that the transformation is not doubly ergodic also works to show it is not weakly doubly ergodic, and another proof is given in [8]. For reference we recall that the Hajian-Kakutani transformation is obtained when at stage $k$ we add $2h_{k}$ subintervals instead of $2h_{k}+1$ subintervals; this transformation has eigenvalues and is not weakly mixing, in fact it is not totally ergodic (its square is not ergodic). In this section we show that the HK(+1) transformation is EIC. We need the following lemma.

Let $T$ denote an invertible ergodic nonsingular transformation on a $\sigma$-finite measure space $(X,\mu)$. Let $\omega_{n}=\omega_{n}^{\mu}=\frac{d\mu\circ T^{n}}{d\mu}$. Define the ratio set

where $B_{\epsilon}(t)=\{x\in[0,\infty):|x-t|<\epsilon\}$. If $\nu$ is a $\sigma$-finite measure equivalent to $\mu$ there exists a positive a.e measurable function $h$ such that $\nu=h\mu$, so $\omega^{\nu}=(h\circ T/h)\omega^{\mu}$, and since every set $A$ of positive measure has a measurable subset of positive measure where $h$ is almost constant, it follows that the ratio set is the same for all equivalent measures. It is well known that $r(T)\setminus\{0\}$ is a multiplicative subgroup of the positive real numbers, see e.g. [9]. If $r(T)\neq\{1\}$, then $T$ admits no equivalent $\sigma$-finite $T$-invariant measure and we say that it is type III. There are several types of type III transformations:

• 1.

  Type $\text{III}_{\lambda}$: $r(T)=\{\lambda^{n}:n\in\mathbb{Z}\}\cup\{0\}$ for some $\lambda\in(0,1)$

• 2.

  Type $\text{III}_{0}$: $r(T)=\{0,1\}$.

• 3.

  Type $\text{III}_{1}$: $r(T)=[0,\infty)$.

In this section, we will extend our construction above to a type $\text{III}_{\lambda}$ example of a system that is EIC but not WDE.

Let $\lambda$ be a real number with $0<\lambda<1$. We will construct $\text{III}_{\lambda}$ nonsingular tower transformations that we will call nonsingular HK$(+1,\lambda)$ transformations, via cutting and stacking as follows:

• 1.

  Start with the unit interval $[0,1)$ as our $0$th column: $C_{0}=\{[0,1)\}$.

• 2.

  The $k+1$th column $C_{k+1}$ is obtained by cutting intervals of $C_{k}$ into two in a way such that the ratio of the measure of the two pieces is $\lambda$, with the left piece being larger than the right piece. Then insert $2h_{k}+1$ spacers on top of the stack. So $r_{k}=2,s(k,0)=0,s(k,1)=2h_{k}+1$, as before, except that now intervals are cut in the uneven ratio $\lambda$.

Let $(X,S_{\lambda},\mathcal{B},\mu)$ denote the rank-one system we constructed above; when clear from the context we will write $S$ for $S_{\lambda}$. We first show that the system is indeed a type $\text{III}_{\lambda}$ system.

We adapt the argument in [4, Theorem 1.5] to the nonsingular setting to show that $(X,S,\mathcal{B},\mu)$ is in fact not weakly doubly ergodic; this argument was used to prove that the infinite measure-preserving HK(+1) transformation is not doubly ergodic but it also yields that is is not weakly doubly ergodic.

Hence, there exists $A$ and $B$ of positive measure such that there does not exist $n$ such that $\mu(S^{n}A\cap A)>0$ and $\mu(S^{n}A\cap B)>0$. This shows that $S$ is not weakly doubly ergodic. Now we show using a similar argument to \threfth1 that $S$ is EIC.

One can also obtain type $\text{III}_{1}$ examples. In our construction, at even times of the inductive construction, choose a fixed $\lambda_{1}$ and at odd times choose a $\lambda_{2}$ such that $\log(\lambda_{1})/\log(\lambda_{2})$ is irrational; this results in a type $\text{III}_{1}$ transformation by a standard argument (see [9]), and our arguments can also be adapted to show these transformations are EIC and not WDE.

As we have mentioned, in [13], Glasner and Weiss asked whether there exists a nonsingular (or infinite measure) transformation that is WM but not EIC. In this section we consider results inspired by this question.

In this section, we first show that transformations admitting isometric factors to locally compact metric spaces are not weakly mixing. Next, we will show that topological rigidity does not imply measure theoretic rigidity, ruling out an approach to proving the equivalence of non-equivalence of EIC and WM by only using topological rigidity. We will then construct an isometric factor for a family of tower transformations. We will then prove using either \threflocallycompact or the equivalence of EUC and WM that a subset of those tower transformations are not weakly mixing. Finally, we’ll use that result and [2] and [17, Chapter 15] to deduce that there exists irrational rotations satisfying certain rigidity conditions. A similar analysis of the latter two processes described can be found in [1] and [7].

Definition 5.1, as well as Proposition LABEL:locallycompact were introduced in unpublished undergraduate thesis [14] for the case of sigma-finite subinvariant transformations (that include infinite measure-preserving transformations). We state the proposition for the setting of nonsingular transformations and include a proof essentially following the ideas in [14].

Let $(M,\rho)$ be a complete separable metric space and $T:M\to M$ a homeomorphism. If $(q_{i})$ is a topological rigidity sequence for $T$, then it is well known that for any finite measure $\mu$ such that $\mu T=\mu$, $(q_{i})$ is a measurable rigidity sequence for $T$. If $M$ is compact, then there exists a finite invariant measure, but in general, $T$ need not admit a finite invariant measure. Thus, if one finds a topological dynamical system $(M,T,\rho)$ that admits a topological rigidity sequence $(q_{i})$ that is not a rigidity sequence for an irrational rotation, $T$ could still fail to be weakly mixing for any nonsingular measure $\nu$ on $M$. The below example illustrates that fact.