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Convergence of Dynamics and the Perron-Frobenius Operator

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by a uniform mixing-like property of the system.

It is well-known that some mixing properties of a dynamical system are described by the asymptotic behavior of the powers of the associated Perron-Frobenius operator. We recall two examples from [3,Thm 4.4.2]: Firstly, the measure preserving dynamical system (Ω, Σ, µ; ϕ) is ergodic, meaning that the measure of each set A ∈ Σ such that ϕ −1 (A) equals A up to a nullset is either 0 or 1, if and only if the fixed space of P consists of constant functions only. Since P is a Dunford-Schwartz operator and therefore mean ergodic [5,Thm 8.24], ergodicity of the system is furthermore characterized by strong convergence of the Cesàro averages of P to 1 ⊗ 1, the bi-Markov projection onto the constant functions. Secondly, the system is (strongly) mixing, i.e. lim n→∞ µ(ϕ −n (A) ∩ B) = µ(A)µ(B) for all A, B ∈ Σ, (1.2) if and only if the sequence of powers (P n ) n∈N converges in the weak operator topology to the just mentioned projection 1 ⊗ 1.
In this article we are concerned with strong convergence of the sequence (P n ) n∈N , i.e. convergence with respect to the strong operator topology, and its relation to the asymptotic behavior of the underlying measure preserving dynamical system. It seems that this situation has never yet been characterized by a mixing-like property analogously to (1.2). Such a characterization is provided in Section 2: while it is easy to see that strong convergence of (P n ) n∈N to the rank-one projection 1 ⊗ 1 is equivalent to the fact that the convergence in (1.2) is uniform in A ∈ Σ, we furthermore show that this is in turn equivalent to the formally weaker property that for any B ∈ Σ there exists D ∈ Σ with µ(D) > 0 such that Moreover, this property is also related to exactness of the system, which is defined as follows. For each n ∈ N set Σ n := {ϕ −n (A) : A ∈ Σ}. We define the sub-σalgebra Σ ∞ := n∈N Σ n and denote its completion within Σ by Σ ∞ . The measure preserving system (Ω, Σ, µ; ϕ) is called exact if the measure of each element of Σ ∞ is either 0 or 1. It follows from [3, Thm 3.2.3 and 4.4.2] that in case where the measure preserving dynamical system is bimeasurable, i.e. ϕ −1 (A), ϕ(A) ∈ Σ for every A ∈ Σ, the following assertions are equivalent: The sequence (P n ) n∈N converges strongly to 1 ⊗ 1.
(iii) The system is exact.
Note that the equivalence of (ii) and (iii) follows from [10][ Cor 4.1]. In view of this characterization, we call the Frobenius-Perron operator P exact if assertion (ii) holds.
As a consequence of these equivalences we obtain that any bimeasurable exact system is mixing and thus ergodic. However, if the system (Ω, Σ, µ; ϕ) is not ergodic, the powers (P n ) n∈N can still converge to a projection onto the fixed space of P , which is no longer one-dimensional. In Section 3 we extend the above equivalences to non-ergodic systems by showing that in general strong convergence of (P n ) n∈N is equivalent to setwise convergences of the iterates (ϕ n ) n∈N in the measure algebra.
In order to make the last statement precise, we have to introduce a little more notation. On the σ-algebra Σ we define the equivalence relation and denote by Σ/ ∼ the set of its equivalence classes. Note that Σ/ ∼ is called the measure algebra, which is a complete metric space with respect to the metric d(A, B) := µ(A△B), cf. [1,Thm 1.12.6]. For the sake of simplicity we omit the distinction in notation between a set A ∈ Σ and its equivalence class in Σ/ ∼. Furthermore, we denote by the sub-σ-algebra of ϕ-invariant sets and by Σ inv its completion within Σ. Then clearly Σ inv ⊆ Σ ∞ and therefore Σ inv ⊆ Σ ∞ .
For any A ⊆ Ω we define the minimal invariant superset A * as the intersection of all B ⊆ Ω containing A such that ϕ −1 (B) = B. If ϕ is bimeasurable, then the minimal invariant superset of any A ∈ Σ is automatically measurable, see Lemma 3.5. Let us emphasize that our definition of A * is different from that in the literature, where A * is usually defined as the minimal element of Σ inv /∼ that contains A up to a nullset, cf. [6, p. 21]. It is not difficult to see that both definitions coincide up to a nullset if and only if ϕ preserves nullsets.
Finally, let us denote by E(f |Σ inv ) the conditional expectation of f ∈ L 1 (Ω, Σ, µ) with respect to Σ inv , i.e. the unique element in L 1 (Ω, Σ, µ) that contains a Σ invmeasurable representative and that Our main theorem now reads as follows.
If these equivalent assertions hold, then the occurring limits can be identified as follows: Let us remark that the equivalence of (ii) and (iii) is not new, but is essentially contained in [10] and [2].
It is also worth mentioning that in Theorem 1.1 ϕ is not assumed to preserve nullsets, i.e. modifying A by a nullset will in general change the sequence (ϕ n (A)) and its limit in Σ/∼. Moreover, when starting with a transformation of the measure algebra ϕ * : Σ/ ∼→ Σ/ ∼ that induces an exact Perron-Frobenius operator and possesses bimeasurable liftings ϕ : Ω → Ω, Theorem 1.1 states that the convergence of (ϕ n (A)) does not depend on the choice of the lifting, whereas its limit does. In order to illustrate this phenomenon consider the following simple example: Let us conclude this introduction with a short note on bimeasurability. It was proven in [12], see also [11], that a Borel-measurable function ϕ : X → Y between two Polish spaces X and Y preserves Borel sets if and only if ϕ −1 ({y}) is at most countable for all but at most countably many y ∈ Y . For instance, the transformation ϕ(x) := 2x mod 1 is bimeasurable on [0, 1] with respect to the Borel-σ-algebra. In some situations bimeasurability can be seen as a reasonable substitute for invertibility, cf. [13]. However, if ϕ is bimeasurable and bijective, then P = T −1 and the powers of P can thus not be strongly convergent unless P = I. In fact, if P is strongly convergent, it follows from that P f = f for any f .

Uniform Mixing Properties
In the following we characterize strong convergence of the powers of the Perron-Frobenius operator P by mixing-like properties of the underlying measure preserving dynamical system (Ω, Σ, µ; ϕ). In doing so, we make use of lower bound techniques originally invented by Lasota and Yorke in [9,8] to provide a sufficient condition for exactness of P . Ding extended this approach in [4] to characterize strong convergence of the sequence (P n ) n∈N in general. He proved that the limit lim n→∞ P n f exists for any f ∈ L 1 (Ω, Σ, µ) if and only if for each 0 where we use the notation g = g + − g − for the positive and negative part of a function g ∈ L 1 . This work has not yet gotten the attention it deserves and it acts as the key to the results in this section. We also refer to [7] for a detailed discussion and further generalizations of lower bound techniques. We start with a characterization of strong convergence of (P n ) n∈N in general.
for each n ∈ N, assertion (ii) follows.
Next we characterize exactness of P by a mixing-like properties of the dynamical system. Theorem 2.2. Let (Ω, Σ, µ; ϕ) be a measure preserving dynamical system with associated Perron-Frobenius operator P ∈ L (L 1 (Ω, Σ, µ)). Then the following assertions are equivalent: (i) The operator P is exact, i.e. lim P n f = Ω f dµ · 1 for each f ∈ L 1 (Ω, Σ, µ).  In particular, we obtain that Now it follows from Proposition 2.1 that (P n ) n∈N converges with respect to the strong operator topology.
In order to prove that P is exact, it remains to show that the fixed space of P is one-dimensional and thus consists of constant functions only. In view of [3,Thm 4.4.2] this is equivalent to ergodicity of the system. Let Ω 1 ∈ Σ be an invariant set, i.e. ϕ −1 (Ω 1 ) ∼ Ω 1 , and note that Ω 2 := Ω \ Ω 1 is also invariant by [ Therefore, µ(B) = 1 and thus µ(Ω 2 ) = 0. Now assume that D ∩ B is a nullset. Since A := D is contained in Ω 2 up to a nullset, we obtain from assertion (iii) that Since µ(A) > 0 this implies that µ(B) = 0 and thus µ(Ω 2 ) = 1. In both cases we have that µ(Ω 1 ) ∈ {0, 1} which proves that the system is ergodic. Therefore, (P n ) n∈N converges in the strong operator topology to 1 ⊗ 1.  Proof. First note that for every A ∈ Σ it follows from ϕ −1 (ϕ(A)) ⊇ A that µ(ϕ(A)) = µ(ϕ −1 (ϕ(A))) ≥ µ(A). Therefore, the limit of the monotone sequence µ(ϕ m (A)) m∈N always exists. Now assume that the system is exact and let A ∈ Σ. If µ(ϕ n (A)) = 0 for all n ∈ N 0 , then assertion (2.2) is trivial. Otherwise, we have lim µ(ϕ n (A)) = 1 by [3, Thm 3.2.3] and therefore as n tends to infinity.
Conversely, assume that (2.2) holds. Fix A ∈ Σ, set a := lim µ(ϕ m (A)) and let B n := Ω \ ϕ n (A). Then as n tends to infinity. Hence, either a = 0 and consequently A is a nullset by the preliminary note, or a = 1. By [3,Thm 3.2.3], this shows that the system is exact

Convergence of Dynamics
In this section we prove our main result, Theorem 1.1, which is prepared by a series of lemmas and preliminary remarks.
Remark 3.1. Let us recall that the Cesàro averages of the Perron-Frobenius operator P corresponding to a measure preserving dynamical system (Ω, Σ, µ; ϕ) always converge strongly to the conditional expectation E( · |Σ inv ) = E( · |Σ inv ), see e.g. [5,Rem 13.24]. This also implies that the Perron-Frobenius operator and the Koopman operator on L 1 (Ω, Σ, µ) share the same set of fixed points, namely L 1 (Ω, Σ inv , µ). Proof. Let A ∈ Σ and fix a Σ inv -measurable representative of E(1 A |Σ inv ). Then Since the right-hand side does not depend on n ∈ N 0 , it follows that B P m 1 A dµ = 0 for all m ∈ N 0 and therefore Remark 3.3. Let us briefly recall the well-known fact that for a measure preserving dynamical system (Ω, Σ, µ; ϕ) we have Indeed, every A ∈ Σ inv coincides with some set in Σ inv up to a nullset and hence satisfies ϕ −1 (A) ∼ A. Conversely, given a set A ∈ Σ such that A ∼ ϕ −1 (A), it can easily be checked that which implies that A ∈ Σ inv .
As mentioned in the introduction, we always have Σ inv ⊆ Σ ∞ , whereas the converse inclusion does not hold in general. In fact, Σ ∞ = Σ whenever ϕ is bimeasurable and bijective. The next lemma gives a sufficient condition for Σ inv = Σ ∞ to hold. Lemma 3.4. Let (Ω, Σ, µ; ϕ) be a bimeasurable measure preserving dynamical system such that the sequence (ϕ n (A)) converges in Σ/∼. Then Σ ∞ ⊆ Σ inv .
The next lemma will be used to identify the limit of the sequence (ϕ n (A)) n∈N in Σ/∼. Beside this, it shows measurability of minimal invariant supersets.
Lemma 3.5. Let (Ω, Σ, µ; ϕ) be a bimeasurable measure preserving dynamical system. Then the minimal invariant superset of any A ∈ Σ is itself measurable, i.e. A * ∈ Σ inv , and is given by is ϕ-invariant, i.e. ϕ −1 (D) = D. Clearly, A ⊆ D and D ∈ Σ inv . Now let A ⊆ F ⊆ Ω such that ϕ −1 (F ) = F . Since ϕ n (A) ⊆ ϕ n (F ) = ϕ n (ϕ −n (F )) ⊆ F for every n ∈ N 0 , we know that B ⊆ F . Therefore, ϕ −n (B) ⊆ ϕ −n (F ) = F for every n ∈ N 0 , which implies that D ⊆ F . Since F was arbitrary, this show that D is in fact the minimal invariant superset of A, denoted by A * .
Finally, we may conclude from the measure preservation that ϕ −1 (B) ∼ B which implies that D ∼ B.
One implication of Theorem 1.1 is based on the following results by Lin from [10,Thm 4.3]; see also [2, Thm 2] for a generalization.
Theorem 3.6. Let X be a Banach space and P ∈ L (X) be a contractive linear operator. We denote by T ∈ L (X * ) the adjoint of P and set B X * := {g ∈ X * : g ≤ 1}.
Then for any f ∈ X the following are equivalent: (i) lim P n f = 0.
(ii) f, h = 0 for all h ∈ F := n∈N T n B X * .
Now we turn to the proof of Theorem 1.1. (iii) ⇒ (ii): Assume that Σ ∞ ⊆ Σ inv . Let f ∈ L 1 (Ω, Σ, µ) and definef := f − E(f |Σ inv ). By Remark 3.1 we obtain that lim P n f = E(f |Σ inv ) if and only if lim P nf = 0. In view of Theorem 3.6 it thus suffices to show that f , h = 0 for each h ∈ n∈N T n B L ∞ .
So let h ∈ T n B L ∞ . Then for each n ∈ N there exists g n ∈ B L ∞ such that h = g n • ϕ n in the sense of L ∞ . Let us fix Σ-measurable representatives of h and all g n and let A ∈ Σ. Then for every n ∈ N we have h −1 (A) ∼ ϕ −n (g −1 n (A)) ∈ Σ n . Since Σ ∞ ⊆ Σ inv we obtain that Due to the completeness of Σ inv , this already implies that h −1 (A) ∈ Σ inv . We hence showed that any h ∈ T n B L ∞ is Σ inv -measurable. By definition of the conditional expectation we know that for all B ∈ Σ inv . By linearity and density it thus follows that f , h = 0 for all h ∈ n∈N T n B L ∞ . As said in the beginning, assertion (ii) now follows from Theorem 3.6.
In summary, we proved that ϕ n (A) is (up to a nullset) a superset of the support of P n 1 A , which converges to the support of E(1 A |Σ inv ) in Σ/∼. By Lemma 3.2, we also know that A, which is nothing but the support of P 0 1 A , is contained in the support of E(1 A |Σ inv ) up to a nullset. Applying this observation to each of the sets ϕ m (A), we conclude that for any m ∈ N 0 and ε > 0 there exists k ∈ N such that µ ϕ m (A) \ ϕ n (ϕ m (A)) < ε whenever n ≥ k. Now we show that the sequence (ϕ n (A)) n∈N converges to U := By what we have just seen, there exists k ∈ N such that µ ϕ m (A)\ϕ n+m (A) < ε/N for every n ≥ k and all 1 ≤ m ≤ N . Therefore, for all n ≥ k + N . Since clearly ϕ n (A) ⊆ U for every n ∈ N, this proves that lim ϕ n (A) = U in Σ/∼ as claimed. It now follows from Lemma 3.5 that which completes the proof.