Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T08:36:05.243Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalisations of the Euler adic

Published online by Cambridge University Press:  22 October 2010

GABRIEL STRASSER
GABRIEL STRASSER*
Affiliation:
Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090 Vienna. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider generalisations of the so-called Euler adic and investigate dynamical properties like ergodicity and total ergodicity. We prove the existence of a unique fully-supported ergodic measure for these generalisations. We also investigate the structure of non-fully-supported ergodic measures and in addition show that each of these measures (fully- and non-fully-supported) is also totally ergodic. In order to determine these dynamical properties we find closed-form expressions for the generalised Eulerian numbers. Additionally we extend a result given by Frick and Petersen to a wider class of adic transformations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 241 - 256
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adams, T. and Petersen, K.Binomial-coefficient multiples of irrationals. Monatsh. Math 125 (1998), 269–278.CrossRefGoogle Scholar
[2]Bóna, M.Combinatorics of Permutations (Chapman and Hall. 2004).CrossRefGoogle Scholar
[3]Carlitz, L. and Scoville, R.Generalized Eulerian numbers: combinatorial applications. J. Reine Angew. Math. 265 (1974), 110–137.Google Scholar
[4]Franssens, G.On a number pyramid related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles. J. Integer Seq. 9 (2006), Article 06.4.1.Google Scholar
[5]Frick, S. B. Limited scope adic transformations. To appear in Discrete Contin. Dyn. Syst.Google Scholar
[6]Frick, S. B., Keane, M., Petersen, K. and Salama, I.Ergodicity of the adic transformation on the Euler graph. Math. Proc. Camb. Phil. Soc. 141 (2006), 231–238.Google Scholar
[7]Frick, S. B. and Petersen, K.Random permutations and unique fully-supported ergodicity for the Euler adic transformation. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), 876–885.Google Scholar
[8]Frick, S. B. and Petersen, K. Reinforced random walks and adic transformations. To appear in J. Theoret. Probab.Google Scholar
[9]Gnedin, A. and Olshanskii, G.The boundary of the Eulerian number triangle. Mosc. Math. J. 6 (2006), 461–475.Google Scholar
[10]MacMahon, P. A.The divisors of numbers. Proc. London Math. Soc. 19 (1920), 305–340.Google Scholar
[11]Méla, X.A class of nonstationary adic transformations. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), 103–123.CrossRefGoogle Scholar
[12]Petersen, K. and Schmidt, K.Symmetric Gibbs measures. Trans. Amer. Math. Soc. 349 (1997), 2775–2811.CrossRefGoogle Scholar
[13]Petersen, K. and Varchenko, A. The Euler adic dynamical system and path counts in the Euler graph. To appear in Tokyo J. Math.Google Scholar
[14]Sánchez-Peregrino, R.The Lucas congruence for Stirling numbers of the second kind. Acta Arith. 94 (2000), 41–52.CrossRefGoogle Scholar