Affine interval exchange transformations with flips and wandering intervals

There exist uniquely ergodic affine interval exchange transformations of [0,1] with flips having wandering intervals and such that the support of the invariant measure is a Cantor set.


Introduction
Let N be a compact subinterval of either R or the circle S 1 , and let f : N → N be piecewise continuous. We say that a subinterval J ⊂ N is a wandering interval of the map f if the forward iterates f n (J), n = 0, 1, 2, . . . are pairwise disjoint intervals, each not reduced to a point, and the ω-limit set of J is an infinite set.
A great deal of information about the topological dynamics of a map f : N → N is revealed when one knows whether f has wandering intervals. This turns out to be a subtle question whose answer depends on both the topological and regularity properties of the map f .
The question of the existence of wandering intervals first arose when f is a diffeomorphism of the circle S 1 . The Denjoy counterexample shows that even a C 1 diffeomorphism f : S 1 → S 1 may have wandering intervals. This behaviour is ruled out when f is smoother. More specifically, if f is a C 1 diffeomorphism of the circle such that the logarithm of its derivative has bounded variation then f has no wandering intervals [6]. In this case the topological dynamics of f is simple: if f has no periodic points, then f is topologically conjugate to a rotation.
The first results ensuring the absence of wandering intervals on continous maps satisfying some smoothness conditions were provided by Guckenheimer [8], Yoccoz [18], and Blokh and Lyubich [2]. Later on, de Melo et al. [13] generalised these results proving that if N is compact and f : N → N is a C 2 -map with non-flat critical points then f has no wandering intervals. Concerning discontinous maps, Berry and Mestel [1] found a condition which excludes wandering intervals in Lorenz maps -interval maps with a single discontinuity. Of course, conservative maps and, in particular, interval exchange transformations, admit no wandering intervals. We consider the following generalisation of interval exchange transformations.
Let 0 ≤ a < b and let {a, b} ⊂ D ⊂ [a, b] be a discrete set containing n points. We say that an injective, continuously differentiable map T : D \ {a, b} are non-removable discontinuities of T . We say that an AIET is oriented if DT > 0, otherwise we say that T has flips. An isometric IET of n subintervals, shortly an n-IET, is an n-AIET satisfying |DT | = 1 everywhere.
Levitt [11] found an example of a non-uniquely ergodic oriented AIET with wandering intervals. Therefore there are Denjoy counterexamples of arbitrary smoothness. Gutierrez and Camelier [4] constructed an AIET with wandering intervals that is semiconjugate to a self-similar IET. The regularity of conjugacies between AIETs and self-similar IETs is examined by Cobo [5] and by Liousse and Marzougui [12]. Recently, Bressaud, Hubert and Maass [3] provided sufficient conditions for a self-similar IET to have an AIET with a wandering interval semiconjugate to it.
In this paper we present an example of a self-similar IET with flips having the particular property that we can apply the main result of the work [3] to obtain a 5-AIET with flips semiconjugate to the referred IET and having densely distributed wandering intervals. The AIET so obtained is uniquely ergodic [16] (see [14,17]) and the support of the invariant measure is a Cantor set.
A few remarks are due in order to place this example in context. The existence of minimal non-uniquely ergodic AIETs with flips and wandering intervals would follow by the same argument of Levitt [11], provided we knew a minimal nonuniquely ergodic IET with flips. However, no example of minimal non-uniquely ergodic IET with flips is known, although it is possible to insert flips in the example of Keane [10] (for oriented IETs) to get a transitive non-uniquely IET with flips having saddle-connections. Computational evaluations indicate that it is impossible to obtain, via Rauzy induction, examples of self-similar 4-IETs with flips meeting the hypotheses of [3], despite this being possible in the case of oriented 4-IETs (see [4,5]). Thus the example we present here is the simplest possible, in the sense that wandering intervals do not occur for AIETs with flips semiconjugate to a self-similar IET, obtained via Rauzy induction, defined on a smaller number of intervals.

Self-similar interval exchange transformations
. . , n. We shall refer to will be called the log-slope-vector and the flips-vector of T , respectively. Notice that T has flips if and only if some coordinate of τ is equal to −1. Let be such that 0 < z 1 < z 2 < . . . < z n < 1; we define the permutation π associated to T as the one that takes i ∈ {1, 2, . . . , n} to π(i) = j if and only if z j = T (( It should be remarked that an AIET E : [a, b] → [a, b] with flips-vector τ ∈ {−1, 1} n and which has the zero vector as the log-slope-vector is an IET (with flipsvector τ ) and conversely. Let J = [c, d] be a proper subinterval of [a, b]. We say that the IET E is self-similar (on J) if there exists an orientation preserving affine map L : Given The Given i = 0, 1, · · · , n, let y i = L −1 (x i ). In this way, the sequence of discontinuities of E is {y 1 , · · · , y n−1 }.
We say that a non-negative matrix is quasi-positive if some power of it is a positive matrix. A non-negative matrix is quasi-positive if and only if it is both irreducible and aperiodic. Let A be an n × n non-negative matrix whose entries are: where N i is the least non-negative integer such that for some y ∈ (y i−1 , y i ) (and therefore for all y ∈ (y i−1 , y i )), E Ni+1 (y) ∈ J. We shall refer to A as the matrix associated to (E, J). Being self-similar, E is also transitive, which implies the quasipositivity of A. Hence, by the Perron-Frobenius Theorem [7], A possesses exactly one probability right eigenvector α ∈ Λ n , where Λ n = {λ = (λ 1 , . . . , λ n ) | λ i > 0, ∀i}.
Moreover, the eigenvalue µ corresponding to α is simple, real and greater than 1 and, also, all other eigenvalues of A have absolute value less than µ. It was proved by Veech [16] (see also [14,17]) that every self-similar IET is minimal and uniquely ergodic. Furthermore, following Rauzy [15], we conclude that

The theorem of Bressaud, Hubert and Maass
Let A ∈ SL n (Z) and let Q[t] be the ring of polynomials with rational coefficients in one variable. We say that two real eigenvalues θ 1 and θ 2 of A are conjugate if there exists an irreducible polynomial f ∈ Q[t] such that f (θ 1 ) = f (θ 2 ) = 0. We say that an AIET T of [0, 1] is semiconjugate (resp. conjugate) to an IET E of J and let A be the matrix associated to (E, J). Let θ 1 be the Perron-Frobenius eigenvalue of A. Assume that A has a real eigenvalue θ 2 such that (1) 1 < θ 2 (< θ 1 ); (2) θ 1 and θ 2 are conjugate. Then there exists an affine interval exchange transformation T of [0, 1] with wandering intervals that is semiconjugate to E.
Proof. This theorem was proved in [3] for oriented IETs. The same proof holds word for word for IETs with flips. In this case, the AIET T inherits its flips from the IET E through the semiconjugacy previously constructed therein.

The interval exchange transformation E
In this section we shall present the IET we shall use to construct the AIET with flips and wandering intervals. We shall need the Rauzy induction [15] to obtain a minimal, self-induced IET whose associated matrix satisfies all the hypotheses of Theorem 1.

Lemma 2.
The map E is self-similar on the interval J = [0, 1/θ 1 ], and A is precisely the matrix associated to (E, J).
As remarked before, since E self-similar, we have that the matrix associated to (E, J) is quasi-positive. In fact, we have that A is the matrix associated to (E, J). To see that, for i ∈ {0, . . . , 5}, let y i = x i /θ 1 be the points of discontinuity for E. Table 2 shows the itinerary I(i) = {I(i) k } Ni k=1 of each interval (y i−1 , y i ), where N i = min {n > 1 : E n+1 ((y i−1 , y i )) ⊂ J} and I(i) k = r if and only if E k ((y i−1 , y i )) ⊂ (x r−1 , x r ).
The number of times that j occurs in I(i), for i, j ∈ {1, . . . , 5}, is precisely A ji and thus A is the matrix associated to the pair (E, J) as required.
Theorem A. There exists a uniquely ergodic affine interval exchange transformation of [0, 1] with flips having wandering intervals and such that the support of the invariant measure is a Cantor set.  Table 2. Itineraries I(i), i ∈ {1, . . . , 5}.
Proof. By construction, the matrix A associated to (E, J) satisfies hypothesis (1) of Theorem 1. The characteristic polynomial p(t) of A can be written as the product of two irreducible polynomials over Q[t]: Thus the eigenvalues θ 1 and θ 2 are zeros of the same irreducible polynomial of degree four and so are conjugate. Hence, A also verifies hypothesis (2) of Theorem 1, which finishes the proof.
Note that for an AIET T , the forward and backward iterates of a wandering interval J form a pairwise disjoint collection of intervals. Moreover, when T is semiconjugate to a transitive IET, as is the case in Theorem A, the α-limit set and ω-limit set of J coincide.