A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature

: It is well known that C 2 -transformation φ of the unit interval into itself with a Markov partition (2.1) π = { I k : k ∈ K } admits φ -invariant density g ( g ≥ 0, (cid:107) g (cid:107) = 1) if: (2.2) | ( φ n ) (cid:48) | ≥ C 1 > 1 for some n ( expanding condition ); (2.3) | φ (cid:48)(cid:48) ( x )/( φ (cid:48) ( y )) 2 | ≤ C 2 < ∞ ( second derivative condition ); and (2.4) # π < ∞ or φ ( I k ) = [0, 1], for each I k ∈ π . If (2.4) is deleted, then the situation dramatically changes. The cause of this fact was elucidated in connection with so-called Adler’s Theorem ([1] and [2]). However after that time in the literature occur claims and opinions concerning the existence of invariant densities and their properties for Markov Maps, which satisfy (2.2), (2.3) and do not satisfy (2.4), revealing unacquaintance with that question. In this note we discuss the problems arising from the mentioned claims and opinions. Some solutions of that problems are given, in a systematic way, on the base of the already published results and by providing appropriate examples. MSC 2010: 28D05, 37A99, 26A18


Introduction
We begin with the celebrated result of [3]. The authors were well aware that their result cannot be extended to expanding transformations with countably many one-to-one pieces in a simple way (see Th. 2, and the comment below on Cond. (17) there). The real task in that period of time was to nd reasonable additional conditions which would guarantee the existence of density invariant under the action of expanding map with countably many one-to-one pieces. Several attempts was made to accomplish that task (for more details see e.g. a review article [4], and also [5], or [6], Sect. 6).
One of the mentioned attempts was published in [7], as Adler's Theorem. Since no proof was given there, the question arose whether it is true [8]. A solution was published in [1], and [2].
After the above two notes and a few other ones, related with them, were published, some further claims and opinions concerning the existence of invariant densities and their lower and upper bounds for Markov Maps appear in the literature.
Those claims and opinions reveal that their authors were unacquainted with the essence of the problem. That problem is rather of delicate nature. It involves, among other things, the so-called measure-theoretic recurrence property.
In this note we clear up, in a systematic way, the essence of the problems with the aid of examples, comments and some published results.

The problems and examples . Existence of invariant densities
We begin with the following De nition 2.1. Let φ : I → I be countably piecewise one-to-one and C , here I ⊂ R denotes an interval. It is called of Markov type if there exists a partition (mod ) π = {I k : k ∈ K} such that: each I k is an interval, φ k := φ |I k is one-to-one and C from I k onto J k = φ(I k ) and the following condition holds: (2.1) The following result follows from Ths. 1, and 2 in [3]: Assume that φ is of Markov type and satis es: Notice that φ in this example has the following defective property: The transformation given in [9] is another example which gives negative answer to the question in Problem 2.3.
Let us now eliminate that defect by imposing the so-called indecomposability condition (see [10]  Proof. Clearly, φ is of Markov type and satis es all the assumptions of Problem 2.6. Nevertheless there is no φ-invariant density. This is so because it is ergodic and has σ-nite absolutely invariant measure concentrated on the whole I (there exists piecewise constant, not integrable, and φ-invariant function).
Notice that φ in this example ful ls even more restrictive condition than Cond. (2.6 The so-called Adler's Theorem asserts that the answer to the question in Problem 2.8 is positive [7]. But no proof is given there. Further, the comments on that theorem in [12] are restricted to a history of the theorem. However, it was noted that Adler's Theorem may not hold in [8].
The counterexamples, published in [1] and [2], disprove Adler's Theorem, i.e. they give negative answer to the question in Problem 2.8.
In the former paper was also proposed a correction of Adler's Theorem. Namely, in the case of bounded interval I the following additional condition was proposed (see Cond. (1.H3) there): (2.10) While in the case of unbounded interval I, it was proposed the following (see Cond. (1.H4) there):

is a sequence of subsets of I, and each Vn is the union of a nite number of I k 's such that Vn
Note that under the assumptions of Problem 2.8 the two Conditions (2.10) and (2.11) are equivalent.
A more e cient than the last two above conditions is the following one: andσ k is de ned in (2.11).
Note that Condition (2.12) is an analogue of the widely known condition from the theory of Markov Chain, the analogue is explained in [13]. Its e ciency is illustrated by examples in ( [13], Ex. 2.1) and in ( [6], Exs. 4.3.1, and 4.3.2). One has also to underline, that Remark 2.9. Condition (2.12) additionally assures aperiodicity but Condition (2.10) does not (see Example 2.13, below).
The role which each of the last three conditions plays in the problem of the existence of invariant density consists in guaranteeing that the needed measure-theoretic recurrence property holds.
Since the transformations given in Example 2.4 and in [9] have global attractors (single point and the Cantor set, respectively), they are without that property. Note also that they do not satisfy the simple Condition (2.10).
On the other hand, it is not easy to decide, without Condition (2.12), whether or not the above mentioned transformations of Examples 4.3.1, or 4.3.2 in [6] have the needed measure-theoretic recurrence property.
Theorems stated in [14] as Theorem 1.2 and, in more abstract setting, as Theorem 1.3 contain the theorem questioned in [8].
There is also given a proof of Th. 1.3 which is incorrect (the thesis of the Lemma 1.5 does not hold, in general). That fact is not noted in [15].
Theorem 2.2 in [16] is a version of Th. 1.3 from [14]. It is stated under the indispensable Condition (2.10). This condition is incorporated, as Condition c), in the de nition of Markov Map (De nition p. 353).
However, in connection with the Assertion c) of that theorem and the opinion on transitivity Assumption contained in Remark 4c) p. 354, here Condition (2.7), one has to raise two questions. The rst question reads: We begin with example of a Markov type map without property (2.7) (see also [10], Example 2.1, Wtransformation): Example 2.11. Let < a < , and then let ψ : I = [ , ] → I be de ned by The interval I is the so-called inessential interval; the remainder two intervals I , I are essential [10]. Transformation ψ restricted to the last two intervals satis es already condition (2.7). Consequently, the invariant density is supported by I ∪ I .
In general, any transformations of M-type can be decomposed into transformations with property (2.7) and an inessential part (see for details [10]  Let (I, F, φ : I → I; dµ) where I is a space, F is a σ-algebra of its subsets, φ transformation with µ invariant measure. Then φ is exact ⇔ for every A ∈ F, lim n→∞ µ(φ n (A)) = . (2.14) Proof. Now we are going to show that χ is not exact. Note rst that dµ = / dx is the unique invariant density. Further χ(I k ) = I ∪ I for k = , and χ(I k ) = I ∪ I for k = , . From these relations it follows that µ(φ n (I k )) = / for every I k , k = , , , , and n = , , , ..., (2.15) and therefore the criterion (2.14) is not ful lled.
Finally, one needs to complete the opinion on Cond. (2.10) expressed in the Remark 4c), p. 354 of the cited book [16]. The authors claim that it can be somewhat weakened but it is certainly not possible to dispense with it altogether if we want to have that Markov Maps necessarily have invariant density.
The essential role played by Cond. (2.10) for the existence of invariant density has been already explained above.
As for the weakening of that condition, it is in general less e cient than Condition ( In the introduction of [19] is noted that in Chapter 7, Section 4 of the book [20], in English, the proof appears to have an error.
Actually the theorem contained in Chapter 7, Section 4 of that book does not hold. This is so because the theorem in question is stated under somewhat less restrictive conditions than that of the so-called Adler's Theorem. Therefore the above mentioned counterexamples in [1], and [2] disprove that theorem as well.
In a review [21] of the book [22], in Polish, the reviewer claims that the proof of the Theorem 1, § 4, Section 7 on p. 164 is not correct.
This problem is clear up in [23]. It turns out that this is the very same problem as that raised in the introduction of [19].
One has to return to the already mentioned above note [19]. At the end of that note is questioned the double inequality of Remark 1 in [24]. That remark states: then the invariant measure is unique and its density is bounded away from .
The author also claims in ( [24], p. 38) that the density g of the unique invariant measure satis es the following double inequality: On the other hand, the authors in ( [19], p. 868) question the above double inequality (2.17).
Remark 2.14. Additionally the authors suggest that the fault is connected with the use of the idea of regularity functional. This is not the case. The regularity functional has been used to get bounds (see e.g. [6], [13], or [18]). However, the bounds of the double inequality are, in general, not constants as in (2.17), but functions (see: the above cited papers or Coroll. 2.21, below).
Next ifñ = in (2.16), then the remark in question is obviously correct.
Finally, in connection with the discussed Remark 1 in ( [24], pp. 37-38), one has to raise two further questions. The rst question is still connected with the problem of the existence of invariant densities for Markov Maps: The second question is connected with the problem of the existence of the lower and upper bounds of invariant density. It is delayed until the second subsection.
As for the question in Problem 2.15, rst note that Cond. (2.16) is essentially more restrictive than Cond. (2.9) thus it is a restrictive version of Adler's Theorem. Nevertheless, the answer to that question is negative too.
It follows from the repeatedly cited counterexamples published in [1] and [2]. More exactly, the de ned in

. Bounds of invariant densities
We begin this subsection with the question announced at the end of the previous one. It can be formulated as follows: Problem 2.16. Assume that φ is of Markov type in the sense of Def. 2.1 which satis es Cond. (2.5).

Does the invariant density satis es the double inequality (2.17)?
Regarding the Problem 2.16. As was above noted, the authors in ( [19], p. 868) question the above double inequality (2.17) in [24] and suggest that the fault is connected with the use of the idea of regularity functional (see Remark 2.14  The proof of the remainder two Parts (b) and (c) is based on the following two corollaries: then there is a constant C > such that g ≥ C .
Proof. This fact is a simple consequence of the assumptions of the previous Corollary 2.21 and the double inequality (2.20) together with (2.21).
The proof of (2.19) consists of two parts. In the rst part it is proved that there exists a unique ψ-invariant density g ψ ; in the second part it is proved that it satis es (2.19).
To prove the existence of g ψ we show that ψ satis es Cond. (2.12). To this end observe that from the inequalitiesσ