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Let $T$ be a measure-preserving transformation of a finite measure space $(X,\mathcal F, \mu)$. Then, Poincaré's recurrence Theorem states that, for every positive measurable subset $A\subseteq X$, there exists some $n\in \mathbb N$ such that, $\mu\left(A\cap T^{-n}A\right)>0$. However, when considering an infinite measure space, that is, $\mu(X)=+\infty$, this theorem may fail. A counterexample is given by:

Let $X=\mathbb R$, $\mathcal F=$ Borel sigma algebra, and $\mu=$ Lebesgue measure. Consider the measure-preserving transformation $T:X\to X$ by $T(x)=x+\sqrt 2$. This system does not satisfy Poincaré's Recurrence Theorem because each point moves infinitely in one direction and never returns to any given set.

Now recall that a measure preserving transformation $T$ on a measure space $(X,\mathcal F, \mu)$ is ergodic if $$A\in \mathcal F,~T^{-1}A=A\pmod\mu\implies \mu(A)=0\text{, or } \mu(X\setminus A)=0.$$ Now the above one is also an example of an ergodic measure preserving transformation which does not satisfy the Poincaré's recurrence Theorem. Additionally, $T$ is a bijection. For a deeper understanding, I would like to construct an ergodic measure preserving transformation on an infinite measure space $(X,\mathcal F, \mu)$ that does satisfy the Poincaré's recurrence Theorem and is a bijection. After extensive searching, I have been unable to find an example of such a transformation, and I find it quite challenging to construct one. Therefore, I am now considering the possibility of proving the converse statement:
Let $(X,\mathcal F, \mu)$ be an infinite measure space. Then, there does not exist an ergodic, measure-preserving, bijective transformation $T$ on $X$ that satisfies Poincaré's Recurrence Theorem.
Any help in solving this would be greatly appreciated. Thank you for your time and assistance!

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    $\begingroup$ You should check the first chapters of the book by Aaronson on infinite ergodic theory, they should provide some examples. Some examples can be constructed using that random walks on the integers are recurrent (for example, the geodesic flow on cyclic covers of hyperbolic closed surfaces). $\endgroup$ Commented Mar 4 at 19:16
  • $\begingroup$ Consider the transformation $Tx=x-\frac{1}{x}$, here $T$ defined everywhere except on $\{0\}$, a measure zero set. $\endgroup$ Commented Mar 4 at 19:34
  • $\begingroup$ @abcdmath if we consider $X=\mathbb R$, then T is not injective here, so $T$ is not a bijection. $\endgroup$ Commented Mar 4 at 20:09
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    $\begingroup$ Your example of an infinite measure preserving transformation for which recurrence fails is not conservative. If an invertible infinite measure preserving transformation is conservative and ergodic, then Poincaré's recurrence holds. Essentially, the only ergodic examples where it doesn't hold are isomorphic to the example you give (a shift right or left). Also, to get a specific example, you can use cutting and stacking. One of the simplest examples is to take the von Neumann-Kakutani transformation and add a stack of intervals every time you cut in half and stack. $\endgroup$ Commented Mar 4 at 23:42
  • $\begingroup$ @TheOtherTerry Thank you so much for your response. I’m having a bit of trouble understanding the phrase 'add a stack of intervals every time you cut in half and stack.' Could you kindly provide further clarification? $\endgroup$ Commented Mar 5 at 6:15

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Here are papers that give examples. Some of these appear in the book by Aaronson. Published in 1963: Kakutani and Parry - Infinite measure preserving transformations with “mixing”.

Originally published in 1970: Hajian and Kakutani - Example of an ergodic measure preserving transformation on an infinite measure space. Last page of this paper gives a rendering of a transformation similar to the von Neumann–Kakutani transformation but with infinite measure added. See picture below. Also, if the intervals used are left-closed and right-open, then the transformation will be a bijection everywhere (although not continuous). Similar to Aaronson, most papers authored or co-authored by C.E. Silva feature examples of infinite $\sigma$-finite measure preserving transformations that are Poincaré recurrent. See https://scholar.google.com/citations?hl=en&user=i2OSovEAAAAJ&view_op=list_works&sortby=pubdate . Infinite measure preserving transformations which satisfy any of the "weak" mixing conditions will be Poincaré recurrent. As you point out, it is still not trivial to write out an explicit example, as was the case for the first weak mixing, non-strong mixing finite measure preserving transformation. This paper shows that it's possible to retain many of the stronger weak mixing properties, when measure is increased slowly as the transformation is defined Adams - Rigidity Sequences of Power Rationally Weakly Mixing Transformations.

Picture from 1970 Hajian–Kakutani example

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  • $\begingroup$ Thank you so much for your explanation. I believe I understand the transformation presented in the picture, and I have a clearer idea of how $T$ is defined on $[0,1]$. However, I’m still a bit confused about where the infinite measure is used in this context. Could you kindly clarify that part for me? I would really appreciate your help $\endgroup$ Commented Mar 6 at 9:52
  • $\begingroup$ The total measure used in the transformation is infinite because the towers on the right side are growing exponentially in height. $\mu(B_1)=2(\frac{1}{2})=1, \mu(B_2)=8(\frac{1}{4})=2, \mu(B_3)=32(\frac{1}{8})=4, \ldots$. To ensure the transformation is defined on $\mathbb{R}$, alternate between positive and negative intervals from $\mathbb{R}$ for the sets $B_i$. $\endgroup$ Commented Mar 6 at 15:28
  • $\begingroup$ Thank you so much. I got your point. $\endgroup$ Commented Mar 7 at 17:56
  • $\begingroup$ @TheOtherTerry I am having some difficulty understanding your comment regarding the statement, 'To ensure the transformation is defined on $\mathbb R$, alternate between positive and negative intervals from $\mathbb R$ for the sets $B_i$.' I am sorry if it’s a simple concept, but would you be so kind as to help clarify it for me? I would really appreciate your assistance. Thank you so much. $\endgroup$ Commented Mar 11 at 11:01
  • $\begingroup$ @abcdmath For the transformation $T:\mathbb{R}\to \mathbb{R}$ to be a bijection on $\mathbb{R}$, you can set $B_0=[0,1)$, $B_1=[-1,0)$, $B_2=[1,3)$, $B_3=[-5,-1)$, $B_4=[3,11), \ldots$. Alternatively, the transformation could be defined on $[0,\infty)$ by changing the intervals that define $B_i$. I don't think Hajian and Kakutani were concerned with the precise set where $T$ was defined, since these will be measure theoretically isomorphic. I love the hand drawn picture that appeared in their official publication in 1970. $\endgroup$ Commented Mar 11 at 20:01
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Let me begin with some general background concerning the Poincaré recurrence theorem. Let $(X,m)$ be a standard (Lebesgue, Rokhlin - Lebesgue) probability space, and $T$ be a measure class preserving bijection. A positive measure subset $A\subset X$ is called wandering if all its translates $T^n A$ (with $n\in\mathbb Z$) are pairwise disjoint. The transformation $T$ is called conservative if it has no wandering sets and dissipative otherwise; it is completely dissipative if the whole state space $X$ is the union of the translates of a certain wandering set (a "fundamental domain"). Thus, in this language the Poincaré recurrence is precisely equivalent to conservativity of $T$. Further, there is a unique Hopf decomposition of the state space $X$ as the disjoint union $X=C\cup D$, where both $C$ and $D$ are $T$-invariant, and the respective restrictions of $T$ are conservative and completely dissipative. For simplicity, let us also assume that $T$ is aperiodic. Then the conservative part $C$ is the union of purely non-atomic ergodic components of $T$, and the dissipative part $D$ is the union of the purely atomic ergodic components (i.e., of the ones that consist of a single $T$-orbit). In particular, an ergodic transformation is always conservative, unless its state space consists of a single orbit.

Note that the role of the measure $m$ in the above definitions is quite limited and amounts just to checking whether the measures of certain sets are non-zero. Therefore, the Hopf decomposition remains the same if the measure is replaced with an equivalent one (be it finite or infinite). On the other hand, $T$-invariance of the measure $m$ (or its equivalence to an invariant finite or $\sigma$-finite measure) is completely irrelevant.

You asked about an ergodic transformation with an infinite invariant measure that satisfies the Poincaré recurrence theorem, i.e., is conservative. As explained above, this is the same as an ergodic transformation with an infinite purely non-atomic invariant measure, and there is plenty of them, for instance:

(1) The suspension (special transformation) over any ergodic transformation with a purely non-atomic finite invariant measure and a non-integrable roof function;

(2) The time shift on the bilateral path space of any irreducible 0-recurrent Markov chain with respect to its stationary measure.

EDIT Suspension is really one of the basic constructions of ergodic theory, and it is covered in most standard textbooks. Here we need it just in the discrete case. Given a measurable roof function $r:X\to\mathbb N$ on a measurable base space $X$, put $\widetilde X=\{(x,n): 1\le n\le r(x)\}$, so that the fibre of the natural projection $\widetilde X\to X$ over a point $x\in X$ is the finite "stack" $I_x=\{1,2,\dots,r(x)\}$ . Then any measurable map $T:X\to X$ gives rise to the suspension $\widetilde T:\widetilde X\to\widetilde X$ defined in the following way: if you are on a stack $I_x$ below the roof, then $\widetilde T$ consists in moving one level higher on the same stack, whereas if you are at the roof, then $\widetilde T$ consists in moving to the bottom of the stack $I_{Tx}$ (the picture in the answer of The Other Terry is an example of this construction). The claim in (1) above is a consequence of two simple facts concerning the relationship between the ergodic properties of $T$ and $\widetilde T$: (i) if $m$ is a $T$-invariant measure on $X$, then the restriction of the product of $m$ and the counting measure on $\mathbb N$ to $\widetilde X$ is $\widetilde T$-invariant; (ii) additionally, if $m$ above is ergodic, then $\widetilde m$ is also ergodic.

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  • $\begingroup$ Thank you for your answer. However, I’m having some difficulty understanding the first example. Could you please clarify it for me or provide a bit more detail on how to approach it? $\endgroup$ Commented Mar 10 at 18:11
  • $\begingroup$ Could you be more precise and specify these difficulties? Are you comfortable with the notion of suspension? $\endgroup$ Commented Mar 10 at 19:27
  • $\begingroup$ No, I am not comfortable with the notion of suspension. Could you please clarify little? $\endgroup$ Commented Mar 10 at 21:11
  • $\begingroup$ I have added an edit to my answer $\endgroup$ Commented Mar 13 at 7:12

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