ESCAPE DYNAMICS FOR INTERVAL MAPS

. We study the structure of the escape orbits for a certain class of interval maps. This structure is encoded in the escape transition matrix (cid:98) A f of an interval map f , extending the traditional matrix A f which considers the transition among the Markov subintervals. We show that the escape transition matrix is a topological conjugacy invariant. We then characterize the 0–1 matrices that can be fabricated as escape transition matrices of Markov interval maps f with escape sets. This shows the richness of this class of interval maps.

1. Introduction. In this paper we further investigate the class of open dynamical systems, see [1,3,4,13], that arise from Markov interval maps with non trivial escape set, see [10,11].
For a Markov interval map f : I → I, with non trivial escape set, we introduce a 0-1 matrix A f that encapsulates not only the transition among the Markov intervals of f (which is the usual transition matrix) but also the transitions from the Markov intervals to the escape intervals. This leads us to naturally study the influence of A f in the dynamics of (I, f ). In particular, to study and classify different interval maps, with nontrivial escape sets, whose restrictions to the respective maximal invariant Cantor sets are topological conjugated. The traditional transition matrix A f that encodes the transitions of the Markov subintervals in the partition of the domain of f was called Stefan matrix in [8] and later used as a (Markov) transition matrix, see [12,10,11].
Besides the intrinsic interest in the open dynamical systems, the other motivation to further the study of this escape matrix A f arises from the undesirable nonfaithfulness of the representations ν x of the Toeplitz algebra T A f on the orbits of points x in the underlying open dynamical systems (provided by interval maps with trivial escape sets) that we constructed in [6]. Two interval maps f and g can have the same Markov transition matrix A f = A g but different escape matrices such that the representation ν x of T A f is faithful for the open dynamical system (I, f ) but non-faithful for (I, g). In this paper we characterize the 0-1 matrices X that can be realised by interval maps f (which is non-unique), thus A f = X, and then use this in [7] as one of the key ingredients to successfully find a graph algebra for which ν x is indeed a faithful representation. The construction of these representations shows one clear advantage of considering A f instead of A f .
We leave this representation theory viewpoint as it is and concentrate on the framework of the problems we tackle in this paper. More precisely, in [6] we considered the interval maps f where the orbit of a point x can be in the escape set of f -see [9] -namely, f k (x) ∈ I does not belong to the domain of f for a certain k ∈ N. Besides this, we defined a 0-1 transition matrix A f that captures not only the transition among the n Markov subintervals (the traditional transition matrix A f ) but also the transitions from the Markov subintervals to the m escape subintervals (giving rise to B f ). If we gather the Markov subintervals first and then the escape subintervals, then there exists a permutation matrix P such that where the lower blocks are matrices with all entries equal to zero. With this transition matrix A f defined, we tackle two natural problems. The first one concerns topological conjugacy where we indeed prove that two Markov interval maps f, g are topological conjugated whenever A f = P A g P T for a certain permutation matrix P , which is in principle unrelated to the permutation that appears in Eq. (1).
The second problem we tackle is to characterize the 0-1 matrices in the block shape A B 0 m×n 0 m×m that can actually be realised as an escape transition matrix A f for some Markov interval map f (up to permutation as explained in Eq. (1)). If such realisation exists, then it is highly not unique. The rows of A f which are entirely null detect the positions of the escape sets of f . This is crucial for the study of the dynamics. The related problem, with no escape sets, was studied in [5], and the answer was that for every row, the entries of ones must be filled in consecutive entries, which we called interval type matrix, see Definition 2.5. Our analysis also shows that such realisations f are restrictions of Markov interval maps g without escape sets.
The plan for the rest of the paper is as follows. In Sect. 2 we first review the dynamical systems background for the interval maps with escape sets, emphasizing the notion of escape transition matrix as in Definition 2.3. Then in Proposition 2.8 we prove that the escape transition matrix is a topological conjugacy invariant. With the help of the notion of m-configuration, in Subsect. 3 we characterize the block matrices A B 0 m×n 0 m×m that can be realised as escape transition matrices of interval maps (up to row and column permutations), see the main Theorem 3.7 as well as Propositions 3.4 and 3.5. We also conclude that any such interval map f (with escape points) is a restriction of a certain interval map without escape points (see Corollary 3.10).
2. Dynamics of interval maps with escape sets. Given n ∈ N, let Γ be an ordered set {c 0 , c − 1 , c + 1 , ..., c − n−1 , c + n−1 , c n } of (at most) 2n real numbers such that Given Γ as above, we define the collection of closed intervals C={I 1 , ..., I n }, with We also consider the collection of open intervals {E 1 , ..., E n−1 }, with in such a way that I := [c 0 , c n ] = ∪ n j=1 I j ∪ n−1 j=1 E j . We now consider the interval maps for which we can construct partitions of the interval I as in (2), (3) and (4). The minimal partition C satisfying the Definition 2.1 is denoted by C f . We remark that the Markov property (P2) allows us to encode the transitions between the intervals in the so-called (Markov) transition n × n matrix A f = (a ij ), defined as follows: whereJ denotes the interior of a set J. A map f ∈ M(I) uniquely determines (together with the minimal partition C f = {I 1 , ..., I n }): Note that Ω f is colloquially called the survivor set of f .  Note that Ω f is the set of points that remain in dom(f ) under iteration of f , and is usually called a cookie-cutter set, see [9]. The open set is usually called the escape set. Every point in E f will eventually fall, under iteration of f , into the interior of some interval E j (where f is not defined) and the iteration process ends. We may say that x is in the escape set E f of f if and only if there is for some j, then E j = ∅ and c j is a singular point, either a critical point or a discontinuity point of f .
We will consider the equivalence relation R f defined by The relation R f is a countable equivalence relation in the sense that the equivalence If x ∈ Ω f then R f (x) has a graph structure without a preferable vertex. If x ∈ E f then R f (x) has a natural structure of a rooted tree. The root of R f (x) is the point e (x) with no outgoing edge, so f −1 (e(x)) ∈ dom(f ) but e(x) / ∈ dom(f ). For every y ∈ E f there is a least natural number τ (y) such that f τ (y) (y) / ∈ dom (f ), which means that, f τ (y) (y) ∈ E j , for some j such that E j = ∅. The final escape point, for the orbit of y, is then denoted by e (y) := f τ (y) (y) and the final escape interval index is denoted by ι (y), that is, if f τ (y) (y) ∈ E j then ι (y) = j.
In order to describe symbolically the escape orbits, we extend the symbol space adding a symbol for each escape interval E j , which will represent an end for the symbolic sequence. For each escape interval E j we associate a symbol j to distinguish the symbol associated with the interval partition I j . That is, we consider the symbols ordered by: If E j is not an interval, that is E j = ∅, then there is no symbol j. Moreover, we define The address map ad (see Definition 2.2) is extended to the escape set E f with ad (x) := j ∈ Σ E f if x ∈ E j . Therefore, the address map is defined for all points of the interval I except for the points of the boundary of the subintervals I 1 , ..., I n , see (2) and ( The itinerary of a point x ∈ E f is always a finite word terminating in a symbol j ∈ Σ E f . An admissible escape word is a word occurring as the itinerary of an escape point x ∈ E f . These words are formed by such that a ξiξi+1 = 1 for i = 1, 2, ..., k − 1, and terminating on an escape symbol j. 2.1. The escape transition matrix and topological conjugacy. Let f ∈ M(I). As in the last section, we thus have an index set {1, ..., n} Σ E f which is ordered as in (8) and (9). In order to deal with the possible transitions from Markov transition intervals to escape intervals we define the escape transition matrix A f as follows.
For row and column labeling, the matrix A f respects the order given in (8).
Example 2.4. Let f be the map, see Figure 1, The whereas the matrix A f is as follows (see the order in (8)): Note that if we use the row and column labeling order where A f is the Markov transition matrix of f and B f is the transition matrix from Markov subintervals to the escape subintervals. We write 0 p×q for the p × q matrix with zeros everywhere, whereas 1 p×q is the p × q matrix with ones everywhere. In order to understand the relation between the escape matrices A f and A g of two interval maps f, g, we put forward the following definition in the context of generic 0-1 matrices.
1. The matrix X is said to be of interval type if in every row, the entries equal to 1 are all consecutive (cf. Def. 2 in [5]). 2. If π is a permutation of the rows of an interval type matrix X and P π its permutation matrix, then we say that π preserves interval type if the matrix P π XP T π is an interval type matrix.
We can write the above definition as follows. Let X = (x ij ). The follower set of If no confusion arises, we denote F X (i) by F (i). Then the matrix X is of interval type if and only if the follower set F X (i) is a set of consecutive numbers (for every Lemma 2.6. Let X be an interval type matrix. A permutation π preserves the interval type of X if and only if is a set of consecutive numbers for every i ∈ α X . Proof. Given i ∈ α X the set F X (i) corresponds to the positions of entries equal to 1 in the row i of X. The set π (F X (i)) corresponds to the row π (i) ∈ α PπXP T π of the matrix P π XP π . The matrix P π XP T π is of interval type if and only if for every row s ∈ α PπXP T π of P π XP T π the follower set F PπXP T π (s), is a set of consecutive numbers. Since F PπXP T π (s) = π F X π −1 (s) , setting i = π −1 (s) concludes the proof. Now the following result is straightforward.
Corollary 2.7. Let X be an interval type matrix. If π leaves the followers sets of X invariant then it preserves interval type.
We now derive the relation between the escape transition matrices for topologically conjugated interval maps, by considering an interval type matrix X as an escape matrix. Let f • and g • denote the restrictions of each map to the finite union of the interior of the intervals in the domain of f and g, respectively, see (P1) in Definition 2.1. Let f and g ∈ M(I). Let D 1 , ..., D n f and G 1 , ..., G ng be the intervals in the partition of the domains of f and g, respectively, as in Definition 2.1.
We recall that f • and g • are topologically conjugated if there exists a continuous bijective map φ : Note that escape intervals for f are escape intervals for f • and there is a transition between I i and I j through f if and only if there is a transition between Note that A f and A g are necessarily of interval type.
Proposition 2.8. Let f ∈ M(I f ) and g ∈ M(I g ), for certain intervals I f and I g .
The maps f • , g • are topologically conjugated if and only if there is a permutation π such that A f = P π A g P T π . Proof. Let D 1 , D 2 , ..., D r , ..., D n f +m f be the intervals partitioning I f , including escape intervals, with the order of real line, and G 1 , G 2 , ..., G r , ..., G ng+mg be the intervals partitioning I g . We assume that n f are the Markov subintervals (say D m1 , ..., D mn f ) and m f are the escape intervals for f , and similarly for g.
Suppose that f • and g • are topological conjugated. So there is a bijective and bicontinuous map φ : , consequently n f = n g = n and there is a permutation π ∈ S n+m so that φ( This means that a(f ) ij = 1 implies a(g) π(i)π(j) = 1. The same argument shows . Finally let us see that the permutation π preserves interval type. The image of any • D i , through f , must be the union of intervals whose labeling consist of consecutive natural numbers, that is, there are two natural numbers, r i and t i so that f ( The number r i is the position the block of 1's in the row i and t i is the length of the block. On the other hand, the fact they are consecutive means that is also an interval, and g•φ( is an interval. Consequently, the permutation preserves interval type. Conversely, let us assume that we are given Markov matrices f and g such that their escape matrices A f , A g are conjugated by a permutation matrix P π with permutation π which preserves interval type and the escape state set. Since P π preserves escape states also preserve regular states. The matrices have the same dimension and its dimension is partitioned into m + n (m the number of escape states and n the number of regular states). Next, we show that the permutation matrix determine directly an invertible map φ preserving the topological structure. Each nonzero entry in the permutation matrix is in the entry (i, π (i)). The map φ is defined as the linear map sending the interior of each interval D i to the interior of the interval G π(i) . If D i is a escape interval then the sign of φ on D i is arbitrary.
On the other hand, Let t = π (j) and recall that a(g) π(i)t = a(f ) ij . Since the matrices are conjugated precisely by the permutation π, we obtain g(φ( , for every D i and therefore for every point in the interior of the domain of f . Therefore, by construction, φ is invertible and preserving continuity on the interior of the domain of f . Therefore, if we start the above construction using A g and g, then we obtained the topological conjugacy. This permutation satisfies the referred property (of preserving interval type) since The permutated transition matrix is of interval type and can be associated with a certain interval map g 1 (see the graph g 1 in Fig. 3) which is conjugated to f through a certain φ 1 with g • 1 = (φ 1 ) −1 •f • •φ 1 . The graph of φ 1 can be determined by the permutation matrix P π1 where  This permutation does not preserve interval type: associated with a certain interval map g 2 which is not conjugated to f through φ 2 . This permutation preserves the followers sets (invariant sets for π 3 ):

Example 2.11. Let
The conjugated matrix is, naturally, of interval type  Note that m(i) = m(j) for i = j, as a consequence of Definition 3.1. In view of (3) and (4), an m-configuration provides us a way to order the Markov and escape subintervals of an interval map f as follows: We note that for every i = 1, ..., m, e(i) ± 1 ∈ {m(j)} j=1,...,n so that there exists j i = 1, ..., n such that e(i)−1 = m(j i ) (thus e(i)+1 = m(j i +1). From the viewpoint of symbolic dynamics, this is the natural order for the rows and columns of the escape transition matrix A f in Definition 2.3, see also (8). When m = n − 1, there is only one m configuration of n + m, with e(i) = 2i for i = 1, ..., m, m(j) = 2j − 1 for j = 1, ..., n and n + m = 2n − 1. If m = 0, there is no configuration (there is no e(i) and we set m(j) = j for all j = 1, ..., n).
Every m-configuration C m,n of n+m gives rise to a choice of the relative locations of the m escape subintervals in n Markov subintervals of I, as in (14), but it does not give the points in (2) neither the interval map f . However, if such map exists, then the escape symbols are precisely see (9) and, the interval map f is such that with P the permutation matrix determined by the following permutation π Cn,m := 1 2 · · · n n + 1 n + 2 · · · n + m m(1) m(2) · · · m(n) e(1) e(2) · · · e(m) .
Of course, given an interval map f , then the ordering (8) naturally gives rise to an m-configuration of n + m and therefore the above permutation matrix P can be denoted by P f . Note that for every reducible non-negative matrix X there is a permutation matrix P so that where each X ii , i = 1, ..., n is either an irreducible block matrix or an 1 × 1 matrix equal to 0, see [2]. If X is itself irreducible we have X = X 11 and in this case P = I. We then say that P XP T is a normal form of X and we denote it by N P (X) or simply by N (X). This form is not necessarily unique. For example, if the dimension of the irreducible block X 11 is equal to n 1 greater than 1 we have n 1 ! different ways of representing X in irreducible blocks, considering the permutations which fix all the lines and columns except the lines and columns of the block X 11 . We will then have the following equivalent representation Different interval maps f and g may give rise to different permutation matrices P f and P g , but with the same normal form N P f ( A f ) = N Pg ( A g ). In this case, A f = A g and B f = B g . Now we summarize in the following result what we discussed so far. Then Proof. Given such an interval map f , (8) gives rise to an m-configuration C n,m of n + m. The row and columns labeling of A f is (14) whereas that of A f B f 0 0 is given by m(1), ..., m(n), e(1), ..., e(m). Now if we consider the permutation matrix P associated to C n,m as in (16), then we have P A f P T = A f B f 0 0 . If we fix the labeling of the matrix A f , then the normal form of A f is unique (note that A f is primitive because we assume that f is a Markov map, see Definition 2.1). Given m, n ∈ N such that 0 ≤ m < n, we now study the conditions over pairs of matrices A and B with sizes n × n and n × m (respectively) such that there exists an interval map f for which: We will simply write N ( We note that if we fix an m-configuration C m,n , and thus a permutation matrix P Cm,n as in (16), then we may find a matrix A such that N P Cm,n ( A) = A B 0 m×n 0 m×m , but this does not guarantee the existence of a solution for (17) as we will see in the sequel.
• The case m = 0 (so Σ E f = ∅) was answered positively in [5,Proposition 6] for any primitive interval type matrix A. We briefly explain the construction of an interval map f such that A f = A and Σ E f = ∅. Let λ A be the Perron eigenvalue of A and µ A = (v 1 , ..., v n ) the corresponding Perron eigenvector normalised so that  • We now recall here [6, Example 3.7] that realizes the above 2-configuration of 5 (that was considered in [6, Example 3.6]). The underlying interval map is The cases of realisability of 1-configurations of n + 1 from interval maps is of particular interest in the sequel as well. Given (u 1 , ..., u n ) ∈ {0, 1} n , we look for the existence of interval maps f such that for some A (which is the Markov transition matrix A f of f ). To keep it simpler, we work with e(1) = 2 (and so m(1) = 1, m(2) = 3, ..., m(n) = n + 1).
Let now s be the least integer so that u s = 1 and u i = 0 for i < s. If k i2 = 0 with i = 1, 3, 4, ..., n + 1 then k ij = 0 for every j, 1 ≤ j ≤ s and k ij = 1 for every j, s + 1 ≤ j ≤ n + 1.
This implies that K 2 > 0, that is, every entry of K 2 is positive and therefore K is a primitive matrix. Indeed, if the row i of K is a row of 1's then the clearly (K 2 ) ij > 1 since every column of K is non-zero. If the row i of K is in the form 0...01...1 (s is the position of the last 0) then the entry (K 2 ) ij has the term K i,s+1 + K s+1,j = 1. Therefore (K 2 ) ij ≥ 1 for all i, j.
We thus obtain an interval type matrix K such that K 2 > 0. Let λ K be the Perron-Frobenius eigenvalue of K and µ = (µ 1 , ..., µ n+1 ) the corresponding Perron-Frobenius eigenvector normalised so that [5,Proposition 6] shows that there is a piecewise linear map g ∈ PL(I) with slope λ K (which is greater than 1 because K is primitive) such that A g = K and J 1 = [0, c 1 ], ..., J i = [c i−1 , c i ], ..., J n+1 = [c n , 1] is the Markov partition of g (so g| Ji is a piecewise linear map such that g(J i ) = j b ij J j ). Now, we define E 1 :=J 2 , I 1 := J 1 , I 2 := J 3 ,..., I n := J n+1 , and f := g| ∪ n i=1 Ii . Then E 1 is the only escape subinterval of f so that E f = E 1 , and f does fulfill the requested property that there is a transition from I 1 to E 1 if u 1 = 1, and for i > 1, there is a transition from I i to E 1 if and only if u i+1 = 1.
Note that if all the u i 's are one in the last proof, then we get the matrix K as a particular case (with m = 1) of Proposition 3.4.
We illustrate the above constructive proof.
For any 0-1 matrix K = (k ij ), let K i = {j : k ij = 1} (using the associated calligraphic letter). If we have a 0-1 matrix A B 0 m×n 0 m×m of type (n + m) × (n + m) and an m-configuration C m,n of m + n, then for every i = 1, ..., n we set where A i is the follower set as defined in Eq. (13). In that case, A and B must then be interval type matrices.
Since A is primitive and every column of B is non-zero, the matrix K is also primitive because (using the definition of K) (K p ) m(i)m(i ) ≥ (A p ) ii , (K p ) m(i)e(j) ≥ (A p−1 B) ij , (K p ) e(j)m(j ) ≥ 1 and (K p ) e(j)e(j ) ≥ 1 for every p ∈ N, i, i = 1, ..., n and j, j = 1, ..., m. Note that if s such that A s > 0, then the above shows that K s+1 > 0, therefore K is a primitive matrix.
Therefore we can apply [5, Proposition 6] and find a piecewise linear map g ∈ M(I) with slope the Perron-Frobenius eigenvalue λ K of K, the escape set Σ Eg = ∅, and the Markov partition C g = {J 1 , ..., J m+n } such that A g = K. Now we define E e(i)−i =J e(i) and I j = J m(j) for i = 1, ..., m and j = 1, ...n and f = g| ∪ n j=1 Ij . Then it is clear that f does what we want, namely A f = A, A f = K 0 and N ( A f ) = A B 0 m×n 0 m×m as required.
If A and B are interval type matrices, then A B 0 m×n 0 m×m might not be realisable by an escape transition matrix. This depends on the chosen configuration, as the following example shows.
Corollary 3. 10. Let A f be an escape transition matrix of an interval map f . Then f is the restriction of an interval map g with escape set E g = ∅.
Then we consider the matrix K as in the proof of Theorem 3.7, and thus [5] guaranties the existence of an interval map g ∈ M(I) such that A g = K and f is indeed the restriction of g to the union of all the subintervals labelled by m(1), ..., m(n).