Groups Having Elements Conjugate to Their Squares and Connections with Dynamical Systems

ABSTRACT

In recent years, dynamical systems which are conjugate to their squares have been studied in ergodic theory. In this paper we study the consequences of groups having elements which are conjugate to their squares and consider examples arising from topological dynamics and more general dynamical systems

In recent years, dynamical systems which are conjugate to their squares have been studied in ergodic theory. In this paper we study the consequences of groups having elements which are conjugate to their squares and consider examples arising from topological dynamics and more general dynamical systems

Cite this paper

nullG. Goodson, "Groups Having Elements Conjugate to Their Squares and Connections with Dynamical Systems,"*Applied Mathematics*, Vol. 1 No. 5, 2010, pp. 416-424. doi: 10.4236/am.2010.15055.

nullG. Goodson, "Groups Having Elements Conjugate to Their Squares and Connections with Dynamical Systems,"

References

[1] O. N. Ageev, “Spectral Rigidity of Group Actions: Applications to the Case gr ,” Proceedings of the American Mathematical Society, Vol. 134, No. 5, 2005, pp. 1331-1338.

[2] G. R. Goodson, “Ergodic Dynamical Systems Conjugate to Their Composition Squares,” Journal of Acta Mathematica Universitatis Comenianae, Vol. 71, No. 7, 2002, pp. 201-210.

[3] G. R. Goodson, “Spectral Properties of Ergodic Dyna- mical Systems Conjugate to Their Composition Squares,” Colloquium Mathematicum, Vol. 107, 2007, pp. 99-118.

[4] B. Hasselblatt and A. Katok, “A First Course in Dynamical Systems,” Cambridge University Press, Cambridge, 2003.

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[6] A. I. Danilenko, “Weakly Mixing Rank-One Transformations Conjugate to Their Squares,” Studia Mathematica, Vol. 187, No. 1, 2008, pp. 75-93.

[7] B. Marcus, “Unique Ergodicity of the Horocycle Flow: Variable Negative Curvature Case,” Israel Journal of Mathematics, Vol. 21, No. 2-3, 1975, pp. 133-144.

[8] R. L. Devaney, “Chaotic Dynamical Systems,” Benja- min Cummings Publishing Company, California, 1986.

[9] N. J. Fine and G. E. Schweigert, “On the Group of Homeomorphisms of an Arc,” Annals of Mathematics, Vol. 62, No. 2, 1955, pp. 237-253.

[10] M. Kuczma, “On the Functional Equation ,” Annales Polonici Mathematici, Vol. 11, 1961, pp. 161-175.

[11] J. L. King. “The Commutant is the Weak Closure of the Powers, for Rank-One Transformations,” Ergodic Theory and Dynamical Systems, Vol. 6, No. 3, 1983, pp. 363-384.

[12] P. R. Halmos, “Ergodic Theory,” Chelsea Publishing Company, Vermont, 1956.

[13] E. Hewitt and K. A. Ross, “Abstract Harmonic Analysis,” 2nd Edition, Springer-Verlag, Berlin, 1979.

[14] N. Gill, A. G. O’Farell and I. Short. “Reversibility in the Group of Homeomorphisms of the Circle,” Bulletin of the London Mathematical Society, Vol. 41, No. 5, 2009, pp. 885-897.

[1] O. N. Ageev, “Spectral Rigidity of Group Actions: Applications to the Case gr ,” Proceedings of the American Mathematical Society, Vol. 134, No. 5, 2005, pp. 1331-1338.

[2] G. R. Goodson, “Ergodic Dynamical Systems Conjugate to Their Composition Squares,” Journal of Acta Mathematica Universitatis Comenianae, Vol. 71, No. 7, 2002, pp. 201-210.

[3] G. R. Goodson, “Spectral Properties of Ergodic Dyna- mical Systems Conjugate to Their Composition Squares,” Colloquium Mathematicum, Vol. 107, 2007, pp. 99-118.

[4] B. Hasselblatt and A. Katok, “A First Course in Dynamical Systems,” Cambridge University Press, Cambridge, 2003.

[5] P. Walters, “An Introduction to Ergodic Theory,” Springer Verlag, Berlin, 1982.

[6] A. I. Danilenko, “Weakly Mixing Rank-One Transformations Conjugate to Their Squares,” Studia Mathematica, Vol. 187, No. 1, 2008, pp. 75-93.

[7] B. Marcus, “Unique Ergodicity of the Horocycle Flow: Variable Negative Curvature Case,” Israel Journal of Mathematics, Vol. 21, No. 2-3, 1975, pp. 133-144.

[8] R. L. Devaney, “Chaotic Dynamical Systems,” Benja- min Cummings Publishing Company, California, 1986.

[9] N. J. Fine and G. E. Schweigert, “On the Group of Homeomorphisms of an Arc,” Annals of Mathematics, Vol. 62, No. 2, 1955, pp. 237-253.

[10] M. Kuczma, “On the Functional Equation ,” Annales Polonici Mathematici, Vol. 11, 1961, pp. 161-175.

[11] J. L. King. “The Commutant is the Weak Closure of the Powers, for Rank-One Transformations,” Ergodic Theory and Dynamical Systems, Vol. 6, No. 3, 1983, pp. 363-384.

[12] P. R. Halmos, “Ergodic Theory,” Chelsea Publishing Company, Vermont, 1956.

[13] E. Hewitt and K. A. Ross, “Abstract Harmonic Analysis,” 2nd Edition, Springer-Verlag, Berlin, 1979.

[14] N. Gill, A. G. O’Farell and I. Short. “Reversibility in the Group of Homeomorphisms of the Circle,” Bulletin of the London Mathematical Society, Vol. 41, No. 5, 2009, pp. 885-897.