Constant slope maps and the Vere-Jones classification

We study continuous countably piecewise monotone interval maps, and formulate conditions under which these are conjugate to maps of constant slope, particularly when this slope is given by the topological entropy of the map. We confine our investigation to the Markov case and phrase our conditions in the terminology of the Vere-Jones classification of infinite matrices.


Introduction
For a, b ∈ R, a < b, a continuous map T : [a, b] → R is said to be piecewise monotone if there are k ∈ N and points a = c 0 < c 1 < · · · < c k−1 < c k = b such that T is monotone on each [c i , c i+1 ], i = 0, . . . , k − 1. A piecewise monotone map T has constant slope s if |T ′ (x)| = s for all x = c i .
For continuous interval maps with a countably infinite number of pieces of monotonicity neither theorem is true -for examples, see [13] and [5]. One of the few facts that remains true in the countably piecewise monotone setting is: If T is s-Lipschitz then h top (T ) ≤ max{0, log s}.
A continuous interval map T has constant slope s if |T ′ (x)| = s for all but countably many points.
The question we want to address is when a continuous countably piecewise monotone interval map T is conjugate to a map of constant slope λ. Particular attention will be given to the case when a slope is given by the topological entropy of T , which we call linearizability: Definition 1. A continuous map T : [0, 1] → [0, 1] is said to be linearizable if it is conjugate to an interval map of constant slope λ = e htop(T ) .
We will confine ourselves to the Markov case, and explore what can be said if only the transition matrix of a countably piecewise monotone map is known in terms of the Vere-Jones classification [22], refined by [19].
The structure of our paper is as follows.
In Section 2 (CPMM: the class of countably piecewise monotone Markov maps) we make precise the conditions on continuous interval maps under which we conduct our investigation -the set of all such maps will be denoted by CPMM (for countably piecewise monotone Markov). In particular, we introduce a slack countable Markov partition of a map and distinguish between operator, resp. non-operator type.
In Section 3 (Conjugacy of a map from CPMM to a map of constant slope) we rephrase the key equivalence from [3,Theorem 2.5]: for the sake of completeness we formulate Theorem 3, which relates the existence of a conjugacy to an "eigenvalue equation" (5), using both classical and slack countable Markov partitions, see Definition 2.
Section 4 (The Vere-Jones Classification) is devoted to the Vere-Jones classification [22] that we use as a crucial tool in the most of our proofs in later sections.
In Section 5 (Entropy and the Vere-Jones classification in CPMM) we show in Proposition 8 that the topological entropy of a map in question and (the logarithm of) the Perron value of its transition matrix coincide. Using this fact, we are able to verify in Proposition 9 that all the transition matrices of a map corresponding to all the possible Markov partitions of that map belong to the same class in the Vere-Jones classification; so we can speak about the Vere-Jones classification of a map from CPMM.
In Section 6 (Linearizability) we present the main results of this text. We start with Proposition 10 showing two basic properties of a λ-solution of equation (5) and Theorem 7 on leo maps, see Definition 4. Afterwards we describe conditions under which a local window perturbation -Theorems 8, 9, resp. a global window perturbation -Theorems 10, 11 results to a linearizable map. • Two elements of P have pairwise disjoint interiors and [0, 1] \ P is at most countable. • The partition P is finite or countably infinite; • T | i is monotone for each i ∈ P (classical Markov partition) or piecewise monotone for each i ∈ P; in the latter case we will speak of a slack Markov partition. • For every i, j ∈ P and every maximal interval i ′ ⊂ i of monotonicity of T , if T (i ′ ) ∩ j • = ∅, then T (i ′ ) ⊃ j.
Remark 1. The notion of a slack Markov partition will be useful in later sections of this paper where we will work with window perturbations. If #P = ∞, then the ordinal type of P need not be N or Z. Remark 3. Let T : [0, 1] → [0, 1] be a piecewise monotone Markov map, i.e., such that orbits of turning points and endpoints {0, 1} are finite. Those orbits naturally determine a finite Markov partition for T . This partition can be easily be refined, in infinitely ways, to countably infinite Markov partitions. If T is topologically mixing and continuous then we will consider T as an element of CPMM. Proposition 2. Let T ∈ CPMM with a Markov partition P. For every pair i, j ∈ P satisfying T (i) ⊃ j there exist a maximal κ = κ(i, j) ∈ N and intervals i 1 , . . . , i κ ⊂ i with pairwise disjoint interiors such that T | i ℓ is monotone and T (i ℓ ) ⊃ j for each ℓ = 1, . . . , κ.
Proof. Since T ∈ CPMM is topologically mixing, it is not constant on any subinterval of [0, 1]. Fix a pair i, j ∈ P with T (i) ⊃ j. Since T is continuous, there has to be at least one but at most a finite number of pairwise disjoint subintervals of i satisfying the conclusion.
For a given T ∈ CPMM with a Markov partition P, applying Proposition 2 we associate to P the transition matrix M = M (T ) = (m ij ) i,j∈P defined by If P is a classical Markov partition of some T ∈ CPMM then m ij ∈ {0, 1} each i, j ∈ P.
Remark 4. For the sake of clarity we will write (T, P, M ) ∈ CPMM * when a map T ∈ CPMM, a concrete Markov partition P for T and its transition matrix M = M (T ) with respect to P are assumed.
For an infinite matrix M indexed by a countable index set P we can consider the powers M n = (m ij (n)) i,j∈P of M : (i) For each n ∈ N and i, j ∈ P, the entry m ij (n) of M n is finite. Proof. (i) From the continuity of T and the definition of M follows that the sum i∈P m ij is finite for each j ∈ P, which directly implies (i). (ii) For n = 1 this is given by the relation (1) defining the matrix M . The induction step follows from (2) of the product of the nonnegative matrices M and M n−1 .
A matrix M indexed by the elements of P represents a bounded linear operator M on the Banach space ℓ 1 = ℓ 1 (P) of summable sequences indexed by P, provided that the supremum of the columnar sums is finite. Then M is realized through left multiplication The matrix M n represents the nth power M n of M and by Gelfand's formula, the spectral radius r M = lim n→∞ M n 1 n .
Remark 5. If (T, P, M ) ∈ CPMM * the supremum in (3) is finite if and only if Since this condition does not depend on a concrete choice of P, we will say the map T is of an operator type when the condition (4) is fulfilled and of a non-operator type otherwise.

Conjugacy of a map from CPMM to a map of constant slope
This section is devoted to the fundamental observation regarding a possible conjugacy of an element of CPMM to a map of constant slope. It is presented in Theorem 3.
Let (T, P, M ) ∈ CPMM * . We are interested in positive real numbers λ and nonzero nonnegative sequences (v i ) i∈P satisfying M v = λv, or equivalently

Definition 5.
A nonzero nonnegative sequence v = (v i ) i∈P satisfying (5) will be called a λ-solution (for M ). If in addition v ∈ ℓ 1 (P), it will be called a summable λ-solution (for M ).
Remark 6. Since every T ∈ CPMM is topologically mixing, any nonzero nonnegative λ-solution is in fact positive: If v = (v i ) i∈P solves (5), k, j ∈ P and v j > 0 then by Proposition 3(ii) for some sufficiently large n, λ n v k ≥ m kj (n)v j > 0.
Let CPMM λ denote the class of all maps from CPMM of constant slope λ, i.e., S ∈ CPMM λ if |S ′ (x)| = s for all but countably many points.
The core of the following theorem has been proved in [3, Theorem 2.5]. Since we will work with maps from CPMM that are topologically mixing, we use topological conjugacies only -see [1,Proposition 4.6.9]. The theorem will enable us to change freely between classical/slack Markov partitions of the map in question.
Theorem 3. Let T ∈ CPMM. The following conditions are equivalent.
(i) For some λ > 1, the map T is conjugate via a continuous increasing onto map ψ : [0, 1] → [0, 1] to some map S ∈ CPMM λ . (ii) For some classical Markov partition P for T there is a positive summable λ-solution u = (u i ) i∈P of equation (5). (iii) For every classical Markov partition P for T there is a positive summable λ-solution u = (u i ) i∈P of equation (5). (iv) For every slack Markov partition Q for T there is a positive summable λ-

Remark 7.
Recently, Misiurewicz and Roth [15] have pointed out that if v is a λsolution of equation ( Proof of Theorem 3. The equivalence of (i), (ii) and (iii) has been proved in [3]. Since (iv) implies (iii) and (v), it suffices to show that (iii) implies (iv) and (v) implies (ii).
(iii) =⇒ (iv). Let us assume that Q is a slack Markov partition for T . Obviously there is a classical partition P for T which is finer than Q, i.e., every element of P is contained in some element of Q. Using (iii) we can consider a positive summable λ-solution u = (u i ) i∈P of equation (5).
clearly the positive sequence v = (v i ) i∈Q is from ℓ 1 (Q). Denoting (m P ij ) i,j∈P and (m Q ij ) i,j∈Q the transition matrices corresponding to the partitions P, Q, we can write where the equality m Q ik = i ′ ⊂i m P i ′ k ′ follows from the Markov property of T on P and Q: So by (6), for a given slack Markov partition Q (for T ) we find a positive summable λ-solution v = (v i ) i∈Q of equation (5).
(v) =⇒ (ii). Assume that for some slack Markov partition Q for T there is a positive summable λ-solution v = (v i ) i∈Q of equation (5). As in the previous part we can consider a classical Markov partition P finer than Q. Using again property (7) let us put Then u = (u i ′ ) i ′ ∈P is positive and we will show that it is a summable λ-solution of equation (5). Fix an i ′ ∈ P and using the property (7) for j ∈ Q for which T (i ′ ) ⊃ j. Then hence summing (9) through all j's from Q that are T -covered by i ′ ∈ P, we obtain with the help of (8), Since by our assumption on v = (v i ) i∈Q and (8) Maps T ∈ CPMM are continuous, topologically mixing with positive topological entropy. Thus all possible semiconjugacies described in [3, Theorem 2.5] will be in fact conjugacies, see [1, Proposition 4.6.9]. Many properties hold under the assumption of positive entropy or for countably piecewise continuous maps. One interesting example of a countably piecewise continuous and countably piecewise monotone (still topologically mixing) map will be presented in Section 7. However, since the technical details are much more involved and would obscure the ideas, we confine the proofs to CPMM.

The Vere-Jones Classification
Let us consider a matrix M = (m ij ) i,j∈P , where the index set P is finite or countably infinite. The matrix M will be called • irreducible, if for each pair of indices i, j there exists a positive integer n such that m ij (n) > 0, and • aperiodic, if for each index i ∈ P the value gcd{ℓ : m ii (ℓ) > 0} = 1.

Remark 9.
Since T ∈ CPMM is topologically mixing, its transition matrix M is irreducible and aperiodic.
In the sequel we follow the approach suggested by Vere-Jones [22]. (ii) For any value r > 0 and all i, j • the series n m ij (n)r n are either all convergent or all divergent; • as n → ∞, either all or none of the sequences {m ij (n)r n } n tend to zero. In the whole text we will assume that for a given nonnegative irreducible aperiodic matrix M = (m ij ) i,j∈P its Perron value λ M is finite. Since by Proposition 4 the value R = λ −1 M is a common radius of convergence of the power series M ij (z) = n≥0 m ij (n)z n , we immediately obtain for each pair i, j ∈ P, It is well known that in G(M ) • m ij (n) equals to the number of paths of length n connecting i to j.
Following [22], for each n ∈ N we will consider the following coefficients: • First entrance to j: f ij (n) equals the number of paths of length n connecting i to j, without appearance of j in between. • Last exit of i: ℓ ij (n) equals the number of paths of length n connecting i to j, without appearance of i in between.
Clearly f ii (n) = ℓ ii (n) for each i ∈ P. Also it will be useful to introduce • First entrance to P ′ ⊂ P: for a nonempty P ′ ⊂ P and j ∈ P ′ , g P ′ ij (n) equals the number of paths of length n connecting i to j, without appearance of any element of P ′ in between.
The first entrance to P ′ ⊂ P will provide us a new type of a generating function used in (37) and its applications.
Remark 11. Let us denote by Φ ij , Λ ij the radius of convergence of the power Proposition 6 has been stated in [19]. Since the argument showing the part (i) presented in [19] is not correct, we offer our own version of its proof. (i) If there is a vertex j such that R = Φ jj then there exists a strongly connected Proof. For the proof of part (ii) see [19].
Let us prove (i). Fix a vertex j ∈ P for which R = Φ jj and choose arbitrary i = j. We can write where i f jj (n), resp. i f jj (n) denotes the number of f jj -paths of length n that do not contain i, resp. contain i.
Then by our assumption and (11) Let us denote g ij (n) the number of paths of length n connecting i to j, without appearance of i, j after the initial i and before the final j. If we denote 1,j f ii (n) the number of f ii -paths of length n connecting i to i with exactly one appearance of j after the initial i and before the final i, we can write for n ≥ 2 (the coefficients j m ii (n) are defined analogously as j f ii (n) -compare the proof of Theorem 6) By the formula of [1, Lemma 4.3.6] and our assumption (12), for arbitrary i ∈ P \{j} we obtain from (13) either (14) lim sup If (14) is fulfilled for some i the existence of a strongly connected subgraph we get R = Φ ii for each i ∈ P and the conclusion follows from [20, Theorem 2.2]. The assertion (iii) immediately follows from (i) and (ii).
The behavior of the series M ij (z), F ij (z) for z = R was used in the Vere-Jones classification of irreducible aperiodic matrices [22]. Vere-Jones originally distinguished R-transient, null R-recurrent and positive R-recurrent case. Later on, the classification was refined by Ruette in [19], who added strongly positive R-recurrent case. All is summarized in Table 1 which applies independently of the sites i, j ∈ P for M irreducible -compare the last row of Table 1 and Proposition 6. We call corresponding classes of matrices transient, null recurrent, weakly recurrent, strongly recurrent. The last three, resp. two possibilities will occasionally be summarized by 'M is recurrent', resp. 'M is positive recurrent'. Table 1.
There are geometrical criteria -see [20] and also [19] -for cases of the Vere-Jones classification to apply depending on whether the underlying strongly connected directed graph can be enlarged/reduced (in the class of strongly connected directed graphs) without changing the entropy. We will use some of them in Section 7.

Further useful facts.
In the whole paper we are interested in nonzero nonnegative solutions of equation (5). Analogously, in the next proposition we consider nonzero nonnegative subinvariant λ-solutions v = (v i ) i∈P for a matrix M , i.e., satisfying the inequality M v ≤ λv.  (5) proportional to the vector (F ij (R)) i∈P (j ∈ P fixed), A general statement (a slight adaption of [18, Theorem 2]) on solvability of equation (5) is as follows: Proof. Following Chung [9] we will use the analogues of the taboo probabilities: For k ∈ P define k m ij (1) = m ij and for n ≥ 1, clearly, k m ij (n) equals to the number of paths of length n connecting i to j with no appearance of k between. Denote also k m ij (n) = m ij (n) − k m ij (n) the number of paths of length n connecting i to j with at least one appearance of k between. The usual convention that k m ij (0) = δ ij (1 − δ ik ) will be used. The following identities directly follow from the definitions of the corresponding generating functions -see before Table 1 -or are easy to verify: For all i, j, k ∈ P and 0 ≤ z < R, By [18, Theorem 2] the double limit (16) can be replaced by Using the identities (i)-(vi) we can write (17) as The conclusion follows from [18, Theorem 2]. In order to be able to modify nonnegative matrices in question we will need the following observation. In some cases it will enable us to produce transition matrices of maps from CPMM. Let E be the identity matrix, see (2).  (ii) By our assumption, for each i, n ij = 0 except for a finite set of j values, so Theorem 6 and Corollary 1 can be applied. Notice that for any nonnegative v, .  Table 1, M is null, resp. positive recurrent if and only if N is null, resp. positive recurrent.
Finally, let M = (m ij ) i,j∈P be positive recurrent and assume its irreducible submatrix K = (k ij ) i,j∈P ′ for some P ′ ⊂ P, denote L = K + E. Then similarly as above we obtain that λ N = λ M + 1, resp. λ L = λ K + 1. If M is weakly, resp. strongly recurrent, then for some K, resp. for each K we obtain λ N = λ L , resp. λ N > λ L and Theorem 4 can be applied.
This finishes the proof for N = M + E. Now, the case when N = M + ℓE, ℓ > 1, can be verified inductively.

Entropy and the Vere-Jones classification in CPMM
The following statement identifies the topological entropy of a map and the Perron value λ M of its transition matrix. Proof. For the first equality, we start by proving λ M ≤ e htop(T ) . We use Proposition 4(i) and Proposition 3(ii). By those statements, λ M = lim n [m jj (n)] 1 n for any j ∈ P and and for each sufficiently large n, the interval j contains m jj (n) intervals j 1 , . . . , j m jj (n) with pairwise disjoint interiors such that T n (j i ) ⊃ j ℓ for all 1 ≤ i, ℓ ≤ m jj (n). Clearly, the map T n has a m jj (n)-horseshoe [12]  Claim. There are finitely many elements i 1 , . . . , i m ∈ P such that S(ε) ⊂ m j=1 i • j .
Proof of Claim. Let us denote P = [0, 1] \ i∈P i • . Then P is closed, at most countable and T (P ) ⊂ P . Assume that x ∈ P ∩ S(ε) = ∅. Then orb T (x) ⊂ P which is impossible for (S(ε), T | S(ε) ) minimal of positive topological entropy. If S(ε) intersected infinitely many elements of P then, since S(ε) is closed, it would intersect also P , a contradiction. Thus, there are finitely many i 1 , . . . , i m ∈ P of the required property.
Our claim together with Proposition 2 say that connect-the-dots map of (S(ε), T | S(ε) ) is piecewise monotone and the finite submatrix M ′ of M corresponding to the elements i 1 , . . . , i m satisfies r(M ′ ) ≥ e htop(T | S(ε) ) . Now the conclusion follows from Proposition 5.
The second statement follows from Theorem 3, Proposition 1 and the fact that topological entropy is a conjugacy invariant: λ M = e htop(T ) = e htop(S) ≤ λ.
We would like to transfer the Vere-Jones classification to CPMM. That is why it is necessary to be sure that a change of Markov partition for the map in question does not change the Vere-Jones type of its transition matrix. This is guaranteed by the following proposition.
Given T ∈ CPMM, consider the family (P α ) α of all Markov partitions for T . Write First, let us assume that P ⊂ Q. Fix two elements j ∈ P, resp. j ′ ∈ Q such that j ′ ⊂ j. Let us consider a path of the length n (20) j = j 0 → P j 1 → P j 2 · · · → P j n = j with respect to P; by Proposition 2 each interval j i contains k i = k P (j i , j i+1 ) intervals of monotonicity of T -denote them ι i (1), . . . , ι i (k i ) -such that T (x) / ∈ j i+1 whenever x ∈ j i \ k i ℓ=1 ι i . This implies that is the number of paths with respect to P through the same vertices in order given by (20) and, at the same time, it is an upper bound of a number of paths Considering all possible paths in (20) and summing their numbers given by (21), we obtain (22) m Q j ′ j ′ (n) ≤ m P jj (n) for each n. On the other hand, since T is topologically mixing and Markov, there is a positive integer ℓ = ℓ(j, j ′ ) such that T ℓ (j ′ ) ⊃ j. It implies for each n, Using (22) and (23), we can write, Hence by the third row of Table 1, M P is recurrent if and only if M Q is recurrent. Again from (22) and (23) we can see that lim n m P jj (n)λ −n is positive if and only if lim n m Q j ′ j ′ (n)λ −n is positive and the fourth row of Table 1 for R = λ −1 can be applied.
In order to distinguish weak, resp. strong recurrence, for a P ′ ⊂ P let Q ′ ⊂ Q be such that Using (22) and (23)  Remark 12. Let (T, P, M ) ∈ CPMM * . Applying Proposition 9 in what follows we will call T transient, null recurrent, weakly recurrent or strongly recurrent respectively if it is the case for its transition matrix M . The last three, resp. two possibilities will occasionally be summarized by 'T is recurrent', resp. 'T is positive recurrent'. It is well known that if T is piecewise monotone then it is strongly recurrent [23, Theorem 0.16].

Linearizability
In this section we investigate in more details the set of maps from CPMM that are conjugate to maps of constant slope (linearizable, in particular). Relying on Theorem 3, Theorem 5 and Proposition 9 our main tools will be local and global perturbations of maps from CPMM resulting to maps from CPMM. Some examples illustrating the results achieved in this section will be presented in Section 7.
We start with an easy but rather useful observation. Its second part will play the key role in our evaluation using centralized perturbation -formula (37) and its applications. (i) If T is leo then any λ-solution of (5) is summable.
The fundamental conclusion regarding linearizability of a map from CPMM provided by the Vere-Jones theory follows.
Theorem 7. If T ∈ CPMM is leo and recurrent, then T is linearizable.
Proof. By assumption there exists a Markov partition P for T such that the transition matrix M = M (T ) = (m ij ) i,j∈P is recurrent. In such a case equation (5) has a λ M -solution described in Theorem 5. Since T is leo the λ M -solution is summable by Proposition 10(i) and the conclusion follows from Theorem 3.
Remark 13. In Section 7 we present various examples illustrating Theorem 7. In particular, we show a strongly recurrent non-leo map of an operator type that is not conjugate to any map of constant slope.
6.1. Window perturbation. In this subsection we introduce and study two types of perturbations of a map T from CPMM: local and global window perturbation.
Definition 6. For S ∈ CPMM with Markov partition P, let j ∈ P such that S |j is monotone. We say that T ∈ CPMM is a window perturbation of S on j (of order k, k ∈ N), if Notice that due to Definition 6 a window perturbation does not change partition P (but renders it slack). Using a sufficiently fine Markov partition for S, its window perturbation T can be arbitrarily close to S with respect to the supremum norm.
In the above definition an element of monotonicity of a partition is used. So, for example we can take P classical (i.e., non-slack), or to a given partition P ′ and a given maximal interval of monotonicity i of a map we can consider a partition P ′′ finer than P ′ such that i ∈ P ′′ . Proposition 11. Let T ∈ CPMM be a window perturbation of a map S ∈ CPMM. The following is true.
(i) If S is recurrent then T is strongly recurrent and R T < R S . (ii) If S is transient then T is strongly recurrent for each sufficiently large k.
Proof. Fix a partition P for S, let T be a window perturbation of S on j ∈ P. Applying Proposition 9 it is sufficient to specify the Vere-Jones class of T with respect to P. Consider generating functions F S (z) = F S jj (z) = n≥1 f S (n)z n , resp. F T (z) = F T jj (z) = n≥1 f T (n)z n , corresponding to S, resp. T and with radius of the convergence Φ S = Φ S jj , resp. Φ T = Φ T jj . Notice that (25) ∀ n ∈ N : f T (n) = (2k + 1)f S (n), (i) If S is recurrent then by Table 1 and (25), hence R T < Φ T and T is strongly recurrent.
(ii) If S is transient then by Table 1 and (25), If for a sufficiently large k, (2k + 1)s > 1, necessarily R T < R S = Φ S = Φ T and T is strongly recurrent by Table 1.
Let M be a matrix indexed by the elements of some P and representing a bounded linear operator M on the Banach space ℓ 1 = ℓ 1 (P) -see Section 2. It is well known [21, p. 264], [21,Theorem 3.3] that for λ > r M the formula defines the resolvent operator R λ (M) : ℓ 1 (P) → ℓ 1 (P) to the operator We will repeatedly use this fact when proving our main results. The following theorem implies that in the space of maps from CPMM of operator type an arbitrarily small (with respect to the supremum norm) local change of a map will result to a linearizable map.
Theorem 8. Let T ∈ CPMM be a window perturbation of order k of a map S ∈ CPMM of operator type. Then T is linearizable for every sufficiently large k.
Proof. We will use the same notation as in the proof of Proposition 11.
Let us denote M T (k) = (m T (k) ij ) i,j∈P the transition matrix of a considered window perturbation T (k) of S, let λ T (k) be the value ensured for M T (k) by Proposition 4, put R T (k) = 1/λ T (k) . Since S is of operator type, it is also the case for each T (k). Using Proposition 11 and Theorem 5 we obtain that for some k 0 the perturbation T (k 0 ) is recurrent and equation (5) is λ T (k 0 ) -solvable: (n)z n , ℓ ∈ P. Since by (25) for each k, we can deduce that (R T (k) ) k≥1 is decreasing and By our definition of a window perturbation, for each i ∈ P \ {j}, now since T (k) is recurrent, Theorem 5 and Theorem 3 can be applied.
In the corresponding strongly connected directed graph G = G(M ), n(i, j) is the length of the shortest path from i to j. In particular, such a path contains neither i nor j inside, so at the same time ℓ ij (n(i, j)) = 0, f ij (n(i, j)) = 0 and ℓ ij (n) = f ij (n) = 0 for every n < n(i, j). Since for every pair j, j ′ ∈ P, the suprema (29) S(j, P) : = sup i∈P n(j, i) n(i, j) , j ∈ P are either all finite or all infinite. Moreover, we have the following.
Proposition 12. Let T ∈ CPMM with two Markov partitions P, resp. Q. Then S(k, P) is finite for some k ∈ P if and only if S(k ′ , Q) is finite for some k ′ ∈ Q.
Proof. Let P = [0, 1] \ j∈P j • and Q = [0, 1] \ j∈Q j • . First, let us assume that P ⊂ Q. Fix two elements j ∈ P, j ′ ∈ Q such that j ′ ⊂ j. Since the map T is topologically mixing, there exists a positive integer m for which T m j ′ ⊃ j.
For an i ∈ P and an i ′ ∈ Q satisfying i ′ ⊂ i we obtain n(i ′ , j ′ ) ≥ n(i, j) and n(j ′ , i ′ ) ≤ n(j, i) + m; hence Inequality (30) together with property (29) show that if S(k, P) is finite for some k ∈ P then S(k ′ , Q) is finite for some k ′ ∈ Q.
On the other hand, there has to exist an i ′′ ∈ Q, i ′′ ⊂ i such that T n(i,j) i ′′ ⊃ j ′ , i.e., n(i, j) ≥ n(i ′′ , j ′ ). Since also n(j, i) ≤ n(j ′ , i ′′ ), we can write for i ′′ ∈ Q inequality (31) together with property (29) show that if S(k ′ , Q) is finite for some k ′ ∈ Q then S(k, P) is finite for some k ∈ P.
If P Q and Q P , we can consider the partition for T Clearly, R = [0, 1] \ j∈R j • = P ∪ Q and we can use the above arguments for the pairs R, P and R, Q hence the conclusion for the pair P, Q follows.
So, in (29), for fixed (T, P, M ) ∈ CPMM * and j ∈ P, we compare the shortest path from j to i (numerator) to the shortest path from i to j (denominator) and take the supremum with respect to i. For example, for our map from Subsection 7.3 the values (29) are equal to 1, when T is leo, (29) is finite for every j ∈ P. Theorem 8 explains the role of a window perturbation in case of maps of operator type. In Theorem 9 we obtain an analogous statement for maps of non-operator type under the assumption that the quantities in (29) are finite.
Theorem 9. Let S ∈ CPMM with a Markov partition Q and such that the supremum in (29) is finite for some j ′ ∈ Q. Let T ∈ CPMM be a window perturbation of order k of S. Then T is linearizable for every sufficiently large k.
Proof. Fix a partition P for S and j ∈ P. A perturbation of S on j of order k ∈ N will be denoted by T (k). By our assumption, Proposition 12 and (29), the supremum S(j, P) is finite. The numbers n(j, i), n(i, j), i ∈ P, do not depend on any window perturbation on an element of P, because such a perturbation does not change P; we define V (n) = {i ∈ P : n(i, j) = n}, c(n) = max{n(j, i) : i ∈ V (n)}, V (n, p) = {i ∈ V (n) : n(j, i) = p}, 1 ≤ p ≤ c(n). Obviously for every n, (32) c(n) n ≤ sup i∈P n(j, i) n(i, j) = S(j, P) < ∞.
To simplify our notation, using Proposition 11 we will assume that S is strongly recurrent, so this is also true for T (k). Similarly as in the proof of Proposition 11 we obtain for each k, Moreover, as in (27), the sequence (R T (k) ) k≥1 is decreasing and lim k R T (k) = 0, i.e., lim k λ T (k) = ∞.
Let us show that for each sufficiently large k there is a summable λ T (k) -solution v = (v i ) i∈P of equation (5). Using (28), we can write for any ε > 0, sufficiently large n 0 = n 0 (ε) ∈ N and some positive constants K, K ′ , Since by (33) the value λ does not depend on k and lim k λ T (k) = ∞, from (32) follows that for any k > k 1 . Clearly the value given by a finite number of summands is finite, so taking (34), (35) and (36) together, (R T (k) ) = A + (B + 1) ≤ A + 1 + K ′ · n≥n 0 (9/10) n < ∞ whenever k > k 1 . This finishes the proof.
6.1.2. Global window perturbation. Let S be from CPMM. In this part we will consider a perturbation of S with a Markov partition P consisting of infinitely many window perturbations on elements of P (and with independent orders) done due to Definition 6.
For technical reasons we consider also an empty perturbation (T = S) as centralized.
Let T be a global (centralized) perturbation of S on P ′ P, denote Q = P \ P ′ . We can write for j ∈ P ′ where the coefficients g P ′ ij (n) were defined before Remark 11. We use formula (37) to argue in our proofs.
In the next theorem the perturbation T need not be of an operator type.
Theorem 10. Let (S, P, M ) ∈ CPMM * be recurrent and linearizable. Assume that T is a recurrent centralized perturbation of S on P ′ . If there are finitely many elements of P ′ that are S-covered by elements of P \ P ′ , then T is linearizable.
Proof. Let k 1 , . . . , k m be all elements of P ′ that are S-covered by elements of Q = P \ P ′ . Then ∀ k ∈ P ′ : i∈Q n≥1 Here, the last inequality follows from our assumption that the map S is recurrent and linearizable together with Theorem 5 and Theorem 3. Using (37), (38) and Proposition 10(ii) we obtain So by Theorem 5 and Theorem 3 the map T is linearizable.
In the next theorem the perturbation T need not be of operator type.
Theorem 11. Let S ∈ CPMM be of operator type. If the transition matrix M = M (S) represents an operator M of the spectral radius λ S then any centralized recurrent perturbation T of S such that h top (T ) > h top (S) is linearizable. The entropy assumption is always satisfied when S is recurrent.
Proof. Let T be a centralized perturbation of S on P ′ ⊂ P, denote Q = P \ P ′ . From Proposition 8 and our assumption on the topological entropy of S and T we obtain 1/λ T = R T < R S = 1/λ S . We can write for j ∈ P ′ i∈Q F T ij (R T ) = i∈Q n≥1 k∈P ′ \{j} where the last inequality follows from the fact that g P ′ ik (n) ≤ f ik (n) for each k ∈ P ′ and n ∈ N -for the definition of g P ′ ik (n), see before Remark 11. By our assumption, formula (26) represents the resolvent operator R λ (M) for every λ > λ S . In particular, R λ T (M) is a bounded operator on ℓ 1 (P) [21, p. 264], hence with the help of Remark 11 we obtain ∀ k ∈ P : and (39), (40) can be rewritten as because k∈P ′ F T kj (R T ) < ∞ for topologically mixing T by Proposition 10(ii) and Theorem 5. The conclusion follows from Theorem 5 and (41), (42). It was shown in Proposition 11(i) that for a recurrent S we always have h top (T ) > h top (S).
In order to apply Theorem 11 let us consider any map R ∈ CPMM of operator type, fix ε > 0. By Theorem 8 there is a strongly recurrent linearizable map S of operator type for which R − S < ε. Similarly as in (27) we can conclude that the transition matrix of S satisfies the assumption of Theorem 11. By that theorem, any centralized perturbation T (operator/non-operator) of S is linearizable (such a centralized perturbation T can be taken to satisfy R − T < ε).
Clearly M is irreducible but not aperiodic. It has period 2, so we consider only m ii (2n). Obviously, m ii (2n) = 2n n a n b n .
In the following statement we describe a class of maps that are not conjugate to any map of constant slope. In particular they are not linearizable. A rich space of such maps (not only Markov) has been studied by different methods in [14].
Proposition 13. Let a, b, k, ℓ ∈ N, k even and ℓ odd, consider the matrix M (a, b) defined in (43). Then N = kM (a, b) + ℓE is a transition matrix of a non-leo map T from CPMM. The map T is null recurrent and it is not conjugate to any map of constant slope. The matrix N represents an operator N on ℓ 1 (Z) and Proof. Notice that the entries of N away from resp. on the diagonal are even, resp. odd. Draw a (countably piecewise affine, for example) graph of a map T from CPMM for which N is its transition matrix. Since M (a, b) is null recurrent, the matrix N is also null recurrent by Proposition 7. Solving the difference equation one can verify that equation (5) with M = M (a, b) has a λ-solution if and only if λ ≥ λ M = 2 √ ab (this follows also from Corollary 1) and none of these solutions is summable. So by Proposition 7 and Theorem 3, the map T is not conjugate to any map of constant slope.
For some a, b, c ∈ N let M = M (a, b, c) = (m ij ) i,j∈N∪{0} be given by Again, the matrix M is irreducible but not aperiodic. It has period 2, so we consider only the coefficients f 00 (2n), see Subsection 4.1. In order to find a λ-solution for M we can use the difference equation (46)   We will study several global window perturbations of S of the following general form: let a = (a n ) n≥1 be a sequence of odd positive integers and consider a global window perturbation T a of S such that • the window perturbation on i n is of order (a n − 1)/2 (i.e., if a n = 1 we do not perturb S on i n ).
Then using the notation of Section 4 and Remark 11 we can consider generating functions F (z) = F a (z) = F a 00 (z) = n≥1 f a 00 (n)z n corresponding to the element i 0 : f a 00 (n) = f (n) for each n. One can easily verify that (48) f (1) = 1, f (n) = a 1 · · · a n−1 , n ≥ 2.
With the help of Proposition 8 we denote λ = λ a , resp. Φ = Φ a the topological entropy of T = T a , resp. radius of convergence of F a (z); also we put R = R a = 1/λ a .
Then by (48), f (n) = 1 for n ∈ A(ℓ) and f (n) = 3 n−ℓ−1 for each n ≥ ℓ + 1, hence Therefore by Table 1, R < Φ, i.e., h top (T ) = log λ ∈ (log 3, log 4) -for the upper bound, see [1]. This implies that the map T a(0) is strongly recurrent hence by Theorems 5 and 8 also linearizable for any ℓ.  [15] observed that the map T a(1) is not conjugate to any map of constant slope. It can be shown that for each choice of a sequence a = (a n ) n≥1 such that the corresponding T has finite topological entropy the following dichotomy is true: either T is recurrent and then equation (5) has no λ-solution for λ > e h(T ) , or T is transient and then equation (5) does not have any λ-solution.
To make sure that lim x→1 f (x) = 0, we need λ ∈ (0, ∞) to satisfy So, any sequence (a n ) n≥0 such that (52) has a positive finite solution λ leads to the linearizable map T ∈ CPMM λ . One can easily see that P = {i n } n≥0 is a Markov (slack) partition for T as defined in Section 2.
Applying Proposition 2 we associate to P the transition matrix and also the corresponding strongly connected directed graph G = G(M ): In particular, the number of loops of length n from i 0 to itself is f 00 (n) = 1 + 2a n−1 .
We use the rome technique from [2] (see also [6,Section 9.3]) to compute the entropy of this graph: it is the leading root of the equation Figure 2. The Markov graph of T ∈ CPMM λ ; 1 + 2a n indicates the number of edges in G from i 0 to i n .
If we divide this equation by z, then we get f 00 (n)z −n ; from Table 1 follows that the graph G (the matrix M , the map T ) is recurrent for any choice of a sequence (a n ) n≥0 and corresponding finite λ > 0. Proposition 8 and comparing equations (52) and (53) we find that e htop(T ) = λ M = λ.
By Remark 5 the map T is of operator type if and only if sup n a n < ∞. In this case, by Table 1 and Proposition 8, Φ 00 = 1 > 1/2 ≥ R = 1/λ M , so the corresponding map is always strongly recurrent. For the choice a n = a n for some fixed integer a ≥ 2 the map T is of non-operator type. In this case, n≥1 f 00 (n)a −n = ∞, so e −htop(T ) = 1/λ M = R < a −1 = Φ 00 , hence by Table 1 the map T is still strongly recurrent.
We can also take a n = a n−ψn for some sublinearly growing integer sequence (ψ n ) n≥1 chosen such that (53) holds for z = a, i.e., a = 1 + n≥1 a −n + 2a −ψn . In this case, Φ 00 = R and n f (n) 00 R n = 1, and the system is null-recurrent or weakly recurrent (not strongly recurrent) depending on whether n na −ψn is infinite or finite. 7.2.3. Transient non-operator example from [7]. Although up to now all our main results have been formulated and proved in the context of continuous maps, many statements remain true also for countably piecewise monotone Markov interval maps that are countably piecewise continuous. 1 We will present a countably piecewise continuous, countably piecewise monotone example adapted from [7], where it is studied in detail for its thermodynamic properties.
Let (w k ) k≥0 be a strictly decreasing sequence in [0, 1] with w 0 = 1 and lim k w k = 0. We will consider the partition P = {p k } k∈N , where the interval map T is designed to be linear increasing on each interval p k = [w k , w k−1 ) for k ≥ 2, p 1 = [w 1 , w 0 ], T (p k ) = i≥k−1 p i for k ≥ 2 and T (p 1 ) = [0, 1]. With a slight modification of our definition from Section 2, P is a Markov partition for T and T is leo. Let M = M (T ) be the matrix corresponding to P, see below. In order to have constant slope λ, we need to solve the recursive relation  Figure 4. The leo map T ∈ F ⊂ CPMM.
• F is a conjugacy class of maps in CPMM.
• strongly connected directed graph G = G(M ) contains its copy as a proper subgraph [5, Theorem 4.5, Fig. 3], so due to Theorem 4(i), T is transient. • the common topological entropy equals log 9.
• equation (5) has a positive summable λ-solution for each λ ≥ 9 = e htop(T ) . with similar properties as the previous example. Therefore T is conjugate to a map of constant slope λ whenever λ ≥ 9, and also linearizable, since λ = 9 = λ M = e htop(T ) .
We have discussed the fact that K is a transition matrix of a non-leo map T ∈ CPMM with corresponding Markov partition denoted by P. Clearly by Remark 5 K represents a bounded linear operator -denote it by K -on ℓ 1 (P), so T is of operator type. We can conclude that: (i) λ K = 5 = e htop(T ) -Propositions 8, 13.
(iii) T is not conjugate to a map of constant slope (is not linearizable) -Proposition 13. (iv) T is null recurrent -Proposition 13.
(v) Let P ′ be a Markov partition for T , denote K ′ the transition matrix of T with respect to P ′ representing a bounded linear operator K ′ on ℓ 1 (P ′ ). Since ∀ y ∈ (0, 1) : #T −1 (y) = 5, we have λ K ′ = r K ′ = 5 -see (i), Section 2 and Proposition 8. Then by Theorem 11 any recurrent centralized (operator/non-operator) perturbation of T is linearizable. In particular it is true for any local window perturbation of T on some element of P ′ -Proposition 11(i). (vi) Let P ′ be a Markov partition for T which equals P outside of some interval [a, b] ⊂ (0, 1). Let T ′ be a local window perturbation of T on some element of P ′ ; from the previous paragraph (v) follows that T ′ is strongly recurrent and linearizable. Consider a centralized (operator/non-operator) perturbation T ′′ of T ′ on some P ′′ ⊂ P ′ . Then if T ′′ is recurrent it is linearizable by Theorem 10. Otherwise we can use either Theorem 8 (an operator case) or Theorem 9 (non-operator case, S(j, P ′ ) is finite for j ∈ P ′ ) to show that a local window perturbation of T ′′ of a sufficiently large order is linearizable.