July  2011, 29(3): 1197-1204. doi: 10.3934/dcds.2011.29.1197

Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups

1. 

Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, United States

Received  January 2010 Revised  May 2010 Published  November 2010

Assume that $X$ is a hyperbolic basic set for $f:X\to X$. We show new examples of Lie group fibers $G$ for which, in the class of $C^r, r>0,$ $G$-extensions of $f$, those that are transitive are open and dense. The fibers are semidirect products of compact and nilpotent groups.
Citation: Viorel Niţică. Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1197-1204. doi: 10.3934/dcds.2011.29.1197
References:
[1]

D. Z. Djoković, The union of compact subgroups of a connected locally compact group,, Math. Zeitschrift, 158 (1978), 99.  doi: 10.1007/BF01320860.  Google Scholar

[2]

M. Field, I. Melbourne and A. Török, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets,, Ergod. Theory Dynam. Systems, 25 (2005), 517.  doi: 10.1017/S0143385704000355.  Google Scholar

[3]

M. Goto, A theorem on compact semi-simple groups,, J. Math. Soc. Japan, 1 (1949), 270.  doi: 10.2969/jmsj/00130270.  Google Scholar

[4]

M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups,, Graduate Texts in Mathematics, 62 (1979).   Google Scholar

[5]

M. Kuranishi, On everywhere dense embedding of free groups in Lie groups,, Nagoya Math. J., 2 (1951), 63.   Google Scholar

[6]

I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions,, Ergod. Theory Dynam. Systems, 23 (2003), 611.  doi: 10.1017/S0143385702001554.  Google Scholar

[7]

I. Melbourne, V. Niţică and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems,, Annales Henri Poincaré, 6 (2005), 725.  doi: 10.1007/s00023-005-0221-0.  Google Scholar

[8]

I. Melbourne, V. Niţică and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems,, Discrete Contin. Dynam. Systems, 14 (2006), 355.   Google Scholar

[9]

I. Melbourne, V. Niţică and A. Török, Transitivity of Euclidean-type extensions of hyperbolic systems,, Ergod. Theory Dynam. Systems, 29 (2009), 1585.  doi: 10.1017/S0143385708000886.  Google Scholar

[10]

I. Melbourne, V. Niţică and A. Török, Transitivity of Heisenberg group extensions of hyperbolic systems,, to appear in Ergod. Theory Dynam. Systems., ().   Google Scholar

[11]

V. Niţică, Examples of topologically transitive skew-products,, Discrete Contin. Dynam. Systems, 6 (2000), 351.   Google Scholar

[12]

V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms,, Ergod. Theory Dynam. Systems, 25 (2005), 257.  doi: 10.1017/S0143385704000471.  Google Scholar

[13]

V. Niţică and A. Török, An open and dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one,, Topology, 40 (2001), 259.  doi: 10.1016/S0040-9383(99)00060-9.  Google Scholar

[14]

J. Schreier and S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe,, Fundamenta Math., 24 (1935), 302.   Google Scholar

[15]

T.-S. Wu, The union of compact subgroups of an analytic group,, Trans. Amer. Math. Soc., 331 (1992), 869.  doi: 10.2307/2154147.  Google Scholar

show all references

References:
[1]

D. Z. Djoković, The union of compact subgroups of a connected locally compact group,, Math. Zeitschrift, 158 (1978), 99.  doi: 10.1007/BF01320860.  Google Scholar

[2]

M. Field, I. Melbourne and A. Török, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets,, Ergod. Theory Dynam. Systems, 25 (2005), 517.  doi: 10.1017/S0143385704000355.  Google Scholar

[3]

M. Goto, A theorem on compact semi-simple groups,, J. Math. Soc. Japan, 1 (1949), 270.  doi: 10.2969/jmsj/00130270.  Google Scholar

[4]

M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups,, Graduate Texts in Mathematics, 62 (1979).   Google Scholar

[5]

M. Kuranishi, On everywhere dense embedding of free groups in Lie groups,, Nagoya Math. J., 2 (1951), 63.   Google Scholar

[6]

I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions,, Ergod. Theory Dynam. Systems, 23 (2003), 611.  doi: 10.1017/S0143385702001554.  Google Scholar

[7]

I. Melbourne, V. Niţică and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems,, Annales Henri Poincaré, 6 (2005), 725.  doi: 10.1007/s00023-005-0221-0.  Google Scholar

[8]

I. Melbourne, V. Niţică and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems,, Discrete Contin. Dynam. Systems, 14 (2006), 355.   Google Scholar

[9]

I. Melbourne, V. Niţică and A. Török, Transitivity of Euclidean-type extensions of hyperbolic systems,, Ergod. Theory Dynam. Systems, 29 (2009), 1585.  doi: 10.1017/S0143385708000886.  Google Scholar

[10]

I. Melbourne, V. Niţică and A. Török, Transitivity of Heisenberg group extensions of hyperbolic systems,, to appear in Ergod. Theory Dynam. Systems., ().   Google Scholar

[11]

V. Niţică, Examples of topologically transitive skew-products,, Discrete Contin. Dynam. Systems, 6 (2000), 351.   Google Scholar

[12]

V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms,, Ergod. Theory Dynam. Systems, 25 (2005), 257.  doi: 10.1017/S0143385704000471.  Google Scholar

[13]

V. Niţică and A. Török, An open and dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one,, Topology, 40 (2001), 259.  doi: 10.1016/S0040-9383(99)00060-9.  Google Scholar

[14]

J. Schreier and S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe,, Fundamenta Math., 24 (1935), 302.   Google Scholar

[15]

T.-S. Wu, The union of compact subgroups of an analytic group,, Trans. Amer. Math. Soc., 331 (1992), 869.  doi: 10.2307/2154147.  Google Scholar

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