Every transformation is disjoint from almost every non-classical exchange.
A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira . Recurrent train tracks with a single switch which we call non-classical interval exchanges , form a subclass of linear involutions without flips. They are analogs of classical interval exchanges, and are first return maps for non-orientable measured foliations associated to quadratic differentials on Riemann surfaces. We show that every transformation is disjoint from almost every irreducible non-classical interval exchange. In the appendix, we prove that for almost every pair of quadratic differentials with respect to the Masur-Veech measure, the vertical flows are disjoint.
Here, we are interested in a dynamical property called disjointness for irreducible non-classical exchanges, a subclass of linear involutions without flips . Classical interval exchange maps arise as first return maps to intervals transverse to vertical foliations of abelian differentials on Riemann surfaces. Analogously, non-classical exchanges are first return maps to pairs of disjoint parallel transverse intervals to non-orientable foliations associated to quadratic differentials. Briefly, non-classical exchange are invertible piecewise isometries of a pair of disjoint interval decomposed into finitely many pieces, and satisfying some additional assumptions ([1, Definition 2.1]). Equivalently, non-classical exchanges can be pictorially defined as recurrent train tracks with a single switch on a Riemann surface. The definitions are presented in detail in Section 2.
Two measure preserving transformations and are said to be disjoint if the product measure is the only invariant measure for the product transformation with marginals and . It is a way of saying that two measure preserving transformations are different and in particular, implies that they are not isomorphic. It was shown in  that every ergodic transformation is disjoint from almost every classical exchange. Here, we prove the analogous result for non-classical exchanges under a mild technical condition.
Let be -ergodic. Then is disjoint with respect to almost every irreducible non-classical exchange with an orientation preserving band.
Almost every irreducible non-classical interval exchange is uniquely ergodic . See also  for a proof with techniques similar to this paper. As a result of this we obtain the following corollary:
For any uniquely ergodic non-classical exchange and almost every irreducible non-classical exchange with an orientation preserving band, the product transformation is uniquely ergodic.
The results above are false if there are no orientation preserving bands. As will be clear from the definition, if a non-classical exchange has no orientation preserving bands then leaves and invariant, so it is not even minimal.
In classical interval exchanges, an interval is partitioned into subintervals, these subintervals are permuted and re-glued preserving orientation to get an invertible, piecewise order preserving, piecewise isometry of . The widths of the subintervals and the permutation used for gluing completely determine a classical interval exchange. There is a way to draw these maps pictorially:
We draw the original interval horizontally and then thicken it vertically to get two copies and , the top and the bottom intervals. Let be the map that switches the intervals i.e., and . Divide into subintervals with the prescribed widths. Divide also into subintervals but incorporate the permutation to decide the widths. Thus, each subinterval of pairs off by a translation with a subinterval of with the same width whose placement is determined by the permutation. We join the pair of subintervals by a band of uniform width. As an example, Figure 1.0 shows a classical exchange with 2 bands, which is a rotation.
The composition of the map from to given by the vertical flow up from along the bands, followed by , exhibits the classical exchange, as a map from to . The inverse of this exchange is then the map from to given by flowing the other way. Thus, we have an equivalent formulation of the interval exchange as a Lebesgue measure preserving transformation on the disjoint union . We shall use this formulation since non-classical exchanges are defined in a similar way: the intervals and are partitioned into subintervals such that paired off intervals are joined by bands of uniform width, but now we allow bands to go from to and also from to . The non-classical exchange as a map from to itself is defined by flow along the bands followed by . We shall call the bands that go from to and from to orientation reversing, since the exchange restricted to the subintervals given by the ends of such a band is orientation reversing.
A classical exchange is said to be irreducible, if the permutation used in the gluing is irreducible: for all subsets , we have . Pictorially, a reducible classical exchange is a concatenation of two classical exchanges side by side for all widths of the bands, and the dynamics can be studied by restricting to each piece. For technical reasons, the appropriate notion of irreducibility for non-classical exchanges is subtler than the straightforward irreducibility notion above. This was done by Boissy and Lanneau in . Here, we consider only those non-classical exchanges that satisfy the Boissy and Lanneau definition of irreducibility. The precise definitions and details will be provided in Section 2.
For a fixed permutation of a classical exchange, the set of widths normalized to have sum 1, is the standard simplex in , and carries a natural Lebesgue measure. Thus, the full parameter space is a disjoint union of simplices over irreducible permutations of letters. For non-classical exchanges, after fixing the combinatorics, the set of normalized widths satisfies an additional switch condition that the sum of the widths of orientation reversing bands of is equal to the sum of widths of orientation reversing bands of . Thus, it has codimension 1 in the standard simplex, and inherits a natural Lebesgue measure. We shall call it the configuration space associated to the combinatorics. The full parameter space is the disjoint union of the configuration spaces.
Following , we use a criteria of Hahn and Parry that two ergodic transformations on a pair of measures spaces are disjoint if the induced maps on -functions on the spaces are spectrally singular. Our proof follows the proof in  while verifying the individual steps for non-classical exchanges.
A key step in the proof in  are rigidity sequences for classical exchanges. The existence of these sequences follows from a cyclic approximation theorem of Veech . We prove a similar cyclic approximation theorem for non-classical exchanges, and the existence of rigidity sequences follows from it.
As an analog of Theorem 1.1, we prove disjointness of vertical flows for quadratic differentials in the appendix. To be precise, we show:
With respect to the Masur-Veech measure, for almost every pair of quadratic differentials the vertical flows are disjoint.
The proof of Theorem 1.4 is simpler and relies on mixing of the Teichmüller geodesic flow.
1.1. Outline of the paper:
In Section 2, we define non-classical exchanges and explain irreducibility for them. In Section 3, we define Rauzy induction for non-classical exchanges, and state the key result of Boissy and Lanneau  for attractors of the Rauzy diagram. In Section 4, we summarize the results from  for expansions of non-classical exchanges by iterated Rauzy induction. In Section 5, we state and prove the cyclic approximation theorem for non-classical exchanges. In Section 6, we establish the existence of rigidity sequences. In Section 7, we prove total ergodicity for almost every non-classical exchange with an orientation preserving band present. In Section 8, we assemble the ingredients to conclude the proof of the main theorem verbatim from . In the appendix, we prove disjointness of vertical flows for quadratic differentials.
Chaika was supported in part by NSF grant DMS-1004372. Gadre was supported by a Simons Travel Grant. The authors thank the 2011 Park City Math Institute program.
2. Non-classical Interval Exchanges
Let denote an alphabet over letters. In the definition that follows, the set labels the bands. A classical exchange is determined by the widths of the subintervals and bijections and from to the set as follows: In the plane, draw two vertically aligned horizontal copies and of the interval . Call them the top interval and the bottom interval respectively. Let be the map that switches the intervals i.e., and . Subdivide into subintervals with widths from left to right. Subdivide into subintervals with widths from left to right. For each , join the subinterval of to the subinterval of by a band of uniform width . The vertical flow along the bands from to , followed by exhibits the classical exchange as a map from to . Similarly, the inverse of the interval exchange is realized as a map from to itself by flowing reverse along the bands, followed by . For classical exchanges, every band has one end on and the other on .
To define non-classical exchanges, the intervals and are partitioned into subintervals that come in pairs, so that they can be joined by bands with uniform width. In this case however, we require bands that have both ends in , and bands that have both ends in . Such a band shall be called orientation reversing because the non-classical exchange restricted to the subintervals given by the ends of the band is orientation reversing. As before, the exchange as a map is given by flowing along the band away from the subinterval in question, followed by . The Lebesgue measure on is obviously invariant under .
We shall work with labelled non-classical exchanges i.e., there is a bijection from to the set of bands. As defined by , the labeling can be thought of as given by a generalized permutation which is a 2-1 map from the set to . Thus, denote the ends of the band . A generalized permutation is of type where if the set enumerates the subintervals of from left to right and the set enumerates the subintervals of from left to right. A generalized permutation defines a fixed point free involution of by:
The equivalence classes under can be indexed with the elements of and correspond to the bands in our picture. The generalized permutations encoding a non-classical exchange do not arise from a true permutation i.e., it has a positive integer with and a positive integer with . This means that there are orientation reversing bands for and . Following Kerckhoff, we shall call the positions that are rightmost on the intervals and , the critical positions. We let be the union of subintervals at the ends of band .
After a horizontal isotopy collapsing to points, the spine of the picture can be thought of as an abstract train track . In particular, if admits an embedding into an orientable surface as a large train track i.e., with complementary regions that are polygons or once-punctured polygons, then the non-classical exchanges associated to the combinatorics, give measured foliations on belonging to the stratum of quadratic differentials indicated by the complementary regions. For example, Figure 2.0 shows a non-classical exchange on a 4-punctured sphere, in the principle stratum.
A classical exchange is called irreducible if there is no such that . In other words, a reducible classical exchange is a concatenation of two classical exchanges over the subsets and . In terms of our picture, for all widths of the subintervals, the intervals partition into two subintervals and , and the set of bands partition into two sets and such that all bands in have both ends in , and all bands in have both ends in .
For non-classical exchanges, a straightforward combinatorial reduction as above can be considered. However, for technical reasons described in , a broader notion of reducibility becomes necessary. From now on, we shall assume that the non-classical exchanges considered are irreducible in this sense of Boissy and Lanneau. See the end of Section 3 for more details.
3. Rauzy induction
A key technical tool is Rauzy induction, which is simply the first return map to an appropriate pair of subintervals . A precise definition is also given in Section 2.2 of . Here, we describe it in terms of the picture, and focus on the coding of iterations by products of elementary matrices. These matrices describe the induced map on the parameter space.
Iterations of Rauzy induction are analogous to continued fraction expansions. In fact, when a classical exchange has two bands, the expansion is equivalent to the continued fraction expansion of the ratio of their widths.
Let be a non-classical exchange. Let and be the bands in the critical positions with on . First, suppose that . Then we slice as shown in Figure 3.0 till we hit for the first time. The band remains in its critical position, but typically a different band moves into the other critical position. Furthermore, the new width of is . All other widths remain unchanged. If instead , then we slice in the opposite direction. In either case, we get a new non-classical exchange with combinatorics and widths as described above. This operation is called Rauzy induction. Since Rauzy induction is represented pictorially by one band splitting another, we shall simply call it a split. In fact, this is consistent with the notion of a split in the context of the underlying train-tracks. Iterations of Rauzy induction are therefore called splitting sequences.
In the first instance above, let , and in the second, let . Let and denote the copies of in and respectively. Rauzy induction is then the first return map to . Let denote the non-classical exchange induced on . Iteratively, let be the (nested) subintervals of to which splits give the first return map. Let be the non-classical exchange induced on .
In general, not all instances of Rauzy induction are defined. To enumerate:
When neither of the splits are defined.
When then neither of the splits is defined.
When is an orientation reversing band on and is the only orientation reversing band on , then can split but not the other way round i.e., only one of the splits is defined.
Case (1) is ruled out when the non-classical exchange is irreducible and Case (2) represents a set of measure zero. In fact, as shown in Section of , almost surely, iterations of Rauzy induction continue ad infinitum.
3.1. Encoding expansions by matrices:
3.1.1. Description of the parameter space:
Consider the vector space and let be the set of points with non-negative coordinates. Let denote the standard -simplex in with sum of the coordinates equal to 1. An assignment of widths to the bands is a point in . Normalizing the widths so that their sum is 1 restricts us to .
To be consistent with a generalized permutation , any assignment of widths must satisfy the switch condition defined by . We denote the set of such widths normalized to sum 1 by , and un-normalized widths by . Let and be the set of orientation reversing bands incident on and respectively. The points in satisfy the additional constraint:
Thus is the intersection with of a codimension 1 subspace of . For and , let be the midpoint of the edge of joining the vertices and . The subset is the convex hull of the points and for .
There are finitely many generalized permutations of an alphabet over letters, and hence finitely many convex codimension 1 subsets of that could be . We call the configuration spaces. The full parameter space is a disjoint union of the configuration spaces .
Let denote the identity matrix on . For , let be the -matrix with the entry 1 and all other entries 0. After Rauzy induction, the relationship between the old and new width data is expressed by
where the matrix has the form . In the first instance of the split, when , the matrix ; in the second instance of the split, when , the matrix . Thus, in either case the matrix is an elementary matrix, in particular . If is any matrix then in the instance when , the action on by right multiplication by has the effect that the -th column of is replaced by the sum of the -th column and -th column of . We phrase this as: in the split, the -th column adds to the -th column. A similar statement holds when .
3.2. Rauzy diagram
For non-classical exchanges, one constructs an oriented graph similar to the Rauzy diagram for a classical exchange. However, there are some key differences in this context.
Construct an oriented graph as follows: the nodes of the graph are generalized permutations of an alphabet over letters satisfying the conditions in the third paragraph at the beginning of Section 3. We draw an arrow from to , if results from splitting . For each node , there are at most two arrows coming out of it. A splitting sequence gives us a directed path in .
For irreducible classical exchanges, each connected component of the Rauzy diagram is an attractor i.e., any node can be joined to any other node by a directed path. Each component is called a Rauzy class.
The Rauzy diagram for non-classical exchanges is more complicated and need not have such strong recurrence properties. See the examples in the Appendix of  or see Section 10 of . In particular, the straightforward definition of reducibility does not work from the point of view of the Rauzy diagram: there are generalized permutations that are not obviously reducible that can split to an obviously reducible one with positive probability. Hence, the broader definition of reducibility defined by Boissy and Lanneau , becomes necessary. In , they prove that
Theorem 3.1 ( Theorem C ).
Let be the subset of nodes of corresponding to the strongly irreducible generalized permutations. Then is closed under forward iterations of Rauzy induction. Moreover, each connected component of is strongly connected i.e., any node in a connected component of can be connected to any other node in the same component of by a sequence of splits.
For the rest of the paper, the generalized permutations considered will be irreducible in the stronger sense, and in a single attractor of .
In this section, we analyze the expansion by splitting sequences on the parameter space.
4.0.1. Preliminary notation:
Given a matrix with non-negative entries, we define the projectivization as a map from to itself by
where if in coordinates then . This shall be the norm used throughout. The norm is additive on , i.e. for in , .
4.1. Iterations of Rauzy induction:
Let . The non-classical exchanges with generalized permutation are points in the configuration space . Let . As shown in , almost every has an infinite expansion. An infinite expansion determines an infinite directed path in .
A finite directed path in shall be called a stage in the expansion. Let denote the elementary matrix associated to the split . The image of under the projective linear map lies in . The matrix associated to the stage is given by
The set is the set of all whose expansion begins with .
In the expansion of , whenever it is necessary to emphasize the dependence on , we shall denote the nodes by , the configuration spaces defined by by , and the elementary matrices associated to by . Thus, given a stage , the set is precisely the set of all for which for all .
The actual (or un-normalized) widths are given by
The projectivization belongs to . Recall that is the non-classical exchange on subintervals induced by . The widths of the bands in are exactly . Let denote the union of subintervals of given by the ends of band .
For a point , let be the first return time to under the transformation . It follows immediately that is constant on each . So for any , we will denote the return time by . For , consider the finite set
The entry of counts the number of points in that lie in . In other words, the entry of counts the number of times a point in visits under the exchange , before the first return to .
The proof proceeds by induction. For , if does not split some other band in , then . In the same situation, all entries of are zero and the entry is 1, verifying the lemma. If splits in , then on and in fact which is the same as the entry of , again verifying the lemma.
Now suppose that the lemma is true for . If does not split some other band in then and in fact for all . By induction is the same as the entry of , which is the same as the entry of , verifying the lemma in this case.
On the other hand, if splits in , then and in fact . By induction, the right hand side is the sum of the and entry of which is the same as entry of . So the lemma is verified in this case too. ∎
The sets form a nested sequence in , all containing . Let
Let be a probability measure on the disjoint union invariant under the non-classical exchange . Let be the widths assigned by to the bands. [5, Proposition 4.2] shows that if is minimal i.e., orbits of are dense, then the map is a linear homeomorphism from the set of -invariant probability measures onto the set .
Let be the Lebesgue measure on each configuration space induced by the -volume form on it as a codimension 1 submanifold of , normalized so that the total volume of the configuration space is 1.
To estimate the measure of subsets of , we compare the push-forward measure from to to the measure on . The Radon-Nikodym derivative of with respect to is the Jacobian of the restriction . Integrating over the subset gives its measure. Thus, to give quantitative estimates, one needs to understand better.
Suppose is the same as at some stage , and suppose is a finite splitting sequence starting from . If is roughly the same at all points, then the relative probability that follows is roughly equal to the probability that an expansion starts with . We make this precise below:
Suppose is a stage and the associated matrix. For , we say that is -uniformly distorted if for all
The matrix is -distributed if for all , the columns of satisfy
As shown by the analysis of in Section 8 of , when the columns of are -distributed then is -uniformly distorted. The relative probability statement then becomes:
Suppose is -distributed, and let be a finite splitting sequence starting from . Let denote the sequence followed by . Then, there exists constant that depends only on such that
At any stage , we shall say that two quantities if up to a multiplicative constant that depends only on and is independent of they are the same. With this notation:
The main technical theorem is [5, Theorem 1.2] which we reproduce below:
Theorem 4.6 (Uniform Distortion).
Let be a stage. There exists a constant , independent of , such that for almost every , there is some , depending on , such that followed by is -uniformly distorted. Additionally, can be arranged to be the same as .
Theorem 4.7 (Strong Normality).
In almost every expansion, for any finite sequence starting from , there are infinitely many instances in which immediately follows a -distributed stage.
5. Cyclic approximation
Let denotes the Lebesgue measure on . Let . Let be a non-classical exchange.
Theorem 5.1 (Cyclic approximation).
For almost every set of widths , and any small, there is a positive integers , a band , such that for some
for all .
is linear on the set for all .
Moreover, is orientation preserving in .
We simply check that the individual steps in the original proof by Veech [11, Theorem 1.4] hold for non-classical exchanges.
As a consequence of Theorem 4.6, there is a -distributed stage such that the matrix is positive and some band is orientation preserving in .
Given a constant , let be the subset of satisfying . Because is orientation preserving in , the subset is non-empty and . Consider . By Strong Normality, for almost every there is a finite splitting sequence starting and terminating in and depending on such that contains . Hence, the expansion of begins with the concatenation i.e., followed by . Thus, .
Let be the length as a directed path in of the splitting sequence . Corresponding to the sequence , let be the exchange induced on corresponding subintervals of . Assuming is orientation preserving, the widths for satisfy
In other words,
Recall from Lemma 4.1 that the entry of the matrix counts the number of times a point in visits under the exchange , before returning to . Hence, for , we have . Finally, . This implies
For a positive matrix , let
Recalling inequalities 3.1 and 3.2 from , we get
Let be the original widths. Assuming is orientation preserving, we use to get
Rearranging the inequality above, we get
Finally, choose small enough such that and . Then, properties (1)-(4) in the theorem hold with and . ∎
A main point in the proof above is that is orientation preserving in . This implies that has a component each in . This shall turn out to be relevant later in Section 7.
6. Rigidity sequences
Following , a positive integer is a -rigidity time for a non-classical exchange if
In particular, a consequence of Theorem 5.1 is that for every and for almost every there is a -rigidity time.
A sequence of positive integers is a rigidity sequence for a non-classical exchange if
For a sequence of natural numbers , let be the cardinality of . The sequence is density 1 if .
To prove disjointness it suffices to show [2, Remark 9] that
Any sequence of density 1 contains a rigidity sequence for almost every non-classical exchange.
For any we have .
Let be a sequence of natural numbers with density 1. Almost every non-classical exchange has a rigidity sequence in .
The above theorem follows from:
For every there exists such that any sequence of density contains a rigidity sequence for a set of non-classical exchange of measure .
As a corollary to Theorem 6.2, we get
For any irrational we have .
Since almost every non-classical exchange is totally ergodic, cannot be rational. If is an eigenvalue for some irrational then rotation by is a factor of . Then rigidity sequences of are also rigidity sequences for the rotation. For any , it is possible to construct a sequence of density at least which contains no rigidity sequence for rotation by . See the discussion in  following Corollary 5. This implies that for any , finishing the proof. ∎
At any stage in the expansion (with generalized permutation ), there are constants and probability independent of such that there are future stages that are -distributed with and
This is [5, Proposition 10.21]. Let . A consequence of the previous fact is
For almost every the set of for which for which some -distributed stage satisfies has positive density i.e., [2, Lemma 6] holds for non-classical exchanges.
We define a random variable as follows: Let and be the -th and -th instances of -distribution in the expansion of with corresponding matrices and . If
then set . Using Fact 1, we see that is at worst exponentially distributed. More precisely, for , we have .
Also by Fact 1 we have an estimate on conditional probabilities. That is, for each we have that
It follows that for almost every we have
It follows from the limit above that the set of such that the -distributed stages satisfy has positive density. ∎
The probability that a sequence follows a -distributed stage is roughly the same as the probability that a sequence begins with . Suppose terminates in the generalized permutation . Quantitatively, what is needed in the proof in  is an estimate of the form:
which simply follows from Remark 4.3.
Let be a real number. We need an estimate for the number of maximal columns with norm i.e., columns such that over the set of all possible -distributed stages in the expansions of non-classical exchanges. A simple count shows that the number of vectors for which is . However, for non-classical exchanges, there is a restriction coming from the fact that the projective linear maps must map configuration spaces to configuration spaces. Thus, not every with can be a maximal column . [5, Lemma 12.2] implies that for -distributed stages, the contraction of is uniform in all directions. To be precise, [5, Lemma 12.2] implies that the projectivization of a maximal column has to lie in a -neighborhood of in . This gives the estimate for the number of maximal columns with to be .
In the same situation as Fact 3, we want to estimate the -measure of the set of non-classical exchanges given by a -distributed stage i.e., for a -distributed stage terminating in a generalized permutation we want to estimate in terms of . In [5, Section 8], we analyze the Jacobian of the restriction to configuration spaces of a projective linear map . Precisely, we show that
where is a constant that depends on the stage. [5, Lemma 12.2] then implies that for any -distributed stage, the constant has a lower bound that depends only on . Consequently,
Facts 3 and 4 together imply that the set of for which there is a stage in the Rauzy expansion of such that is at most i.e., [2, Lemma 10] holds for non-classical exchanges. Along with the estimate for the set defined in the proof of Theorem 5.1, the above lemma implies that .
From Lemma 6.4 and Fact 2 it follows that the set of for which has a -rigidity time with fixed previous induction steps in has measure at least .
The estimates at the end of Facts 4 and 5 prove Theorem 6.2 by the exact argument in the proof of [2, Corollary 3]. The key point is that the estimates establish that there is a constant independent of so that among the set of nonclassical exchanges that have a rigidity time between and , for the special reason given above, the set of those for which this time is in a set of density in has proportion at most . This follows from Fact 3 which limits how many -distributed matrices can have the largest column sum for a single . ∎
7. Total ergodicity
A measure preserving transformation is said to be totally ergodic if every forward iterate of it is ergodic. To finish the proof of Theorem 1.1, we show:
If has an orientation preserving band then almost every in is totally ergodic.
We first give a sufficient condition for total ergodicity.
A non-classical exchange is totally ergodic with respect to if for any prime , there are arbitrarily good cyclic approximations of height coprime to .
Suppose is ergodic but not totally ergodic. There is a prime such that is not ergodic.
Suppose is not ergodic. By the ergodic decomposition theorem, there exists a invariant set with such that is the support of an ergodic measure of , and any other ergodic measure of assigns measure 0 to . Any translate is invariant and carries the ergodic measure . Hence, by the decomposition theorem, either and coincide or their intersection is empty i.e. either or . Let . Then . If then the set
is invariant and does not have full measure. Hence, is not ergodic. ∎
Before the formal proof of Proposition 7.2 we state the idea, which we make precise by using Lusin’s Theorem. If is ergodic but not totally ergodic then it has a factor by given by identifying points with the ergodic component of they are in. By assumption there exist such that and are arbitrarily close to the identity. Because is arbitrarily close to the identity and is a factor is close to the identity too. But where mod . Because we have is not close to the identity.
Let be a -invariant set that is the support of an ergodic measure as above. Then the translates are all disjoint.
By Lusin’s Theorem, we may assume that contains all but a very small proportion, say for very large , of a subinterval of of size . Choose for the cyclic approximation to be less than . By assumption, we may chose the height of a cyclic approximation to to be such that gcd and . For , call the sets