Geodesic
The absolute of the comb graph
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 125-135

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In the 1970s R. Stanley introduced the comb graph $\mathbb{E}$ whose vertices are indexed by the set of compositions of positive integers and branching reflects the ordering of compositions by inclusion. A. Vershik defined the absolute of a $\mathbb{Z}_+$-graded graph as the set of all ergodic probability central measures on it. We show that the absolute of $\mathbb{E}$ is naturally parametrized by the space $\Omega = \{(\alpha_1, \alpha_2, \dots ) : \alpha_i \ge 0$, $\sum_i \alpha_i \le 1\}$. 
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     author = {P. P. Nikitin},
     title = {The absolute of the comb graph},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a8/}
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P. P. Nikitin. The absolute of the comb graph. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 125-135. http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a8/