Geodesic
On the combinatorial structure of Rauzy graphs
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 57-73.

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Let $S_m^0$ be the set of all irreducible permutations of the numbers $\{1,\dots,m\}$ ($m\ge3$). We define Rauzy induction mappings $a$ and $b$ acting on the set $S_m^0$. For a permutation $\pi\in S_m^0$, denote by $R(\pi)$ the orbit of the permutation $\pi$ under the mappings $a$ and $b$. This orbit can be endowed with the structure of an oriented graph according to the action of the mappings $a$ and $b$ on this set: the edges of this graph belong to one of the two types, $a$ or $b$. We say that the graph $R(\pi)$ is a tree composed of cycles if any simple cycle in this graph consists of edges of the same type. An equivalent formulation of this condition is as follows: a dual graph $R^*(\pi)$ of $R(\pi)$ is a tree. The main result of the paper is as follows: if the graph $R(\pi)$ of a permutation $\pi\in S_m^0$ is a tree composed of cycles, then the set $R(\pi)$ contains a permutation $\pi_0\colon i\mapsto m+1-i$, $i=1,\dots,m$. The converse result is also proved: the graph $R(\pi_0)$ is a tree composed of cycles; in this case, the structure of the graph is explicitly described. 
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     author = {M. B. Dubashinsky},
     title = {On the combinatorial structure of {Rauzy} graphs},
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     pages = {57--73},
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     volume = {277},
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     url = {http://geodesic.mathdoc.fr/item/TM_2012_277_a4/}
}
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M. B. Dubashinsky. On the combinatorial structure of Rauzy graphs. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 57-73. http://geodesic.mathdoc.fr/item/TM_2012_277_a4/

[1] Kontsevich M., Zorich A., “Connected components of the moduli spaces of Abelian differentials with prescribed singularities”, Invent. math., 153:3 (2003), 631–678 | DOI | MR | Zbl

[2] Masur H., “Interval exchange transformations and measured foliations”, Ann. Math. Ser. 2, 115:1 (1982), 169–200 | DOI | MR | Zbl

[3] Rauzy G., “Échanges d'intervalles et transformations induites”, Acta arith., 34:4 (1979), 315–328 | MR | Zbl

[4] Veech W., “Gauss measures for transformations on the space of interval exchange maps”, Ann. Math. Ser. 2, 115:2 (1982), 201–242 | DOI | MR | Zbl

[5] Veech W., “The Teichmüller geodesic flow”, Ann. Math. Ser. 2, 124:3 (1986), 441–530 | DOI | MR | Zbl

[6] Zorich A., “Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents”, Ann. Inst. Fourier, 46:2 (1996), 325–370 | DOI | MR | Zbl