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Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [
However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up works.
Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method, we can prove most of the identities appearing in the literature so far ([
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Figure 2. $ S' $ is a boosted sequence of $ S $ for $ (x, c) $, because $ \text{red} \circ S' $ gives the same result as $ S \circ \text{red} $. Note that $ S' $ may be not unique: in the diagram we show a second boosted sequence of $ S $ for $ (x, c) $, namely $ S" $. It is not necessarily the case that $ (x', c') : = S'(x, c) $ coincides with $ (x", c") : = S"(x, c) $, but merely that their reductions coincide (they must be both $ y' $)
Figure 4. Outline of the proof of connectivity between $ x_1 $ and $ x_2 $, using the labelling method. The sequence $ S $ sends $ y_1 $ to $ y_2 $, however the intervals containing the gray vertices of $ x'_1 $ may not be at their correct place, in order to match with those of $ x'_2 $. The sequence $ S_* $ corrects for this. Thus the boosted sequence $ S'_* S' $ sends $ x'_1 $ to $ x'_2 $, and $ x_1 $ and $ x_2 $ are connected
Figure 8. Left: A reducible permutation in matrix representation; the $ \mathcal{S} $ dynamic acts on the first block with $ L $ and $ R $. Right: A reducible permutation in diagram representation; the $ \mathcal{S} $ dynamics acts on the gray part with $ L $ and $ R $, while leaving the blue part unchanged
Figure 9. Left: an irreducible permutation, $ \sigma = [\, 451263\, ] $. Right: the construction of the cycle structure. Different cycles are in different colours. The length of a cycle or path, defined as the number of top (or bottom) arcs, is thus 2 for red and violet, and 1 for blue. The green arrows are the endpoints of the rank path, which is in blue. As a result, in this example $ \lambda (\sigma ) = (2, 2) $ (for the cycles of color red and violet), $ r(\sigma ) = 1 $ (corresponding to the rank path of color blue), and $ \ell(\sigma ) = 2 $
Figure 10. Left: an example of permutation, $ \sigma = [\, 251478396\, ] $. Right: an example of subset $ I = \{1, 2, 6, 8, 9\} $ (labels are for the bottom endpoints, edges in $ I $ are in blue). There are two crossings, out of the maximal number $ \binom{|I|}{2} = 10 $, thus $ \chi_I = 8 $ in this case, and this set contributes $ (-1)^{|I|+\chi_I} = (-1)^{5+8} = -1 $ to $ A(\sigma ) $
Figure 20. The first column represents the permutation $ \sigma $ and its the cycle invariant. The second column represents the permutation $ \sigma ' $ obtained from the insertion of a double-edge within the arcs $ \alpha $ and $ \beta $ of $ \sigma $ and the third column displays and demonstrates the new invariant of $ \sigma ' $
Figure 23. The permutation $ \sigma _{1, 4, \{(0.1, 0.1), (0.2, 3.1), (0.3, 1.1), (1.1, 4.1), (1.2, 2.1)\}} $. We cannot show the full matrix $ Q $ for such a big example, but we can give one row, for the edge which has the label $ e $ in the drawing. The row $ Q_e $ reads $ (Q_e)_{11, 12, \ldots, 15, 21, \ldots, 25} = (1, 1, 0, 0, 0, \, 0, 0, 1, 1, 1) $
Figure 37. $ (\sigma '_4, \sigma '_2) $ are both permutations with invariant $ (\lambda , r, s\neq 0) $ and $ \tau_2 $ has sign invariant 0. However proposition 19 applied to $ \sigma '_4, \sigma '_2 $ and $ \tau_2 $ implies that $ \sigma '_4 $ should have invariant $ -s $, This is contradictory and thus the case $ \Pi'''_b(\beta) = b_{0, i-1, 1} $ cannot happen
Figure 38.
$ \sigma $ and $ \sigma ' $ are standard of type $ X(r, i) $. The labels of the principal cycle of $ \sigma $ are send to the arcs of the principal cycle of $ \sigma ' $ by the sequence $ R^{-1}B(S)R $. Indeed they are attached to the $ i $th first labels of the rank of $ \tau $ and the labels of the rank are fixed. Thus they are also attached to $ i $th first the labels of the rank of $ \tau' $ which are the arcs of the principal cycle of $ \sigma ' $.
In the figure, we choose $ \Pi_t(\alpha) = t_{0, i, j} $ instead of $ t_{c, i, j} $ for some $ c $ for space-saving purpose
Table 1.
Cycle, rank and sign invariants of the exceptional classes. The sign
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Table 2.
List of invariants
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Table 3.
Modification to the cycle invariant between
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Let
A representation of a combinatorial structure
Outline of the proof of connectivity between
Diagram representations of matchings and permutations, and matrix representation of permutations
Our two main examples of dynamics, in diagram representation. Top: the
The Rauzy dynamics concerning permutations, in matrix representation
Left: A reducible permutation in matrix representation; the
Left: an irreducible permutation,
Left: an example of permutation,
Two consistent labellings
The action on the dynamics (with an
The proof that
The proof that
The gray edges of
Illustration that the edges, inserted within the labels of a reduced permutation, follow those labels when applying the boosted dynamic
Left: a schematic representation of a permutation of type
The insertion of one edge within the arcs
The first line represents the case: Top arc : any cycle. Bottom arc: any cycle. The second line represents the case: Top arc : rank path. Bottom arc: principal cycle
The first column represents the permutation
An example of a shift-irreducible family
The two permutations
The permutation
From left to right: the permutations
Left: a permutation
The base cases of the induction of the proof of proposition 58
Left: a permutation with a
Two families of base permutations with their respective cycle invariant
The permutation resulting from adding a
The two constructed
The scheme to add two different even cycles to a permutation
Two families of
The case
The case
The construction of the proof of the lemma. We have
The construction of the proof of the lemma 73. Both
The case of the cycle 1-shift. We apply proposition 28 on
The case of the cycle jump. We apply proposition 76 on
The case of the cycle 2-shift. We apply proposition 28 on
Description of the structure of configurations in the identity class