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Ergodicity of the adic transformation on the Euler graph

Published online by Cambridge University Press:  28 September 2006

SARAH BAILEY ,
MICHAEL KEANE ,
KARL PETERSEN  and
IBRAHIM A. SALAMA
SARAH BAILEY
Affiliation:
Department of Mathematics, CB 3250, Phillips Hall, University of North Carolina, Chapel Hill, NC 27599, U.S.A. e-mail: [email protected]
MICHAEL KEANE
Affiliation:
Department of Mathematics, Wesleyan University, 265 Church Street, Middletown, CT 06459, U.S.A. e-mail: [email protected]
KARL PETERSEN
Affiliation:
Department of Mathematics, CB 3250, Phillips Hall, University of North Carolina, Chapel Hill, NC 27599, U.S.A. e-mail: [email protected]
IBRAHIM A. SALAMA
Affiliation:
School of Business, North Carolina Central University, P.O. Box 19407, Durham, NC 27707, U.S.A. e-mail: [email protected]
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Abstract

The Euler graph has vertices labelled $(n,k)$ for $n=0,1,2,\ldots$ and $k=0,1,\ldots,n$, with $k+1$ edges from $(n,k)$ to $(n+1,k)$ and $n-k+1$ edges from $(n,k)$ to $(n+1,k+1)$. The number of paths from (0,0) to $(n,k)$ is the Eulerian number $A(n,k)$, the number of permutations of $1,2,\ldots,n+1$ with exactly $n-k$ falls and k rises. We prove that the adic (Bratteli–Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 141 , Issue 2 , September 2006 , pp. 231 - 238
Copyright
2006 Cambridge Philosophical Society

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