# American Institute of Mathematical Sciences

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. [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. [3] M. Goto, A theorem on compact semi-simple groups,, J. Math. Soc. Japan, 1 (1949), 270. doi: 10.2969/jmsj/00130270. [4] M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups,, Graduate Texts in Mathematics, 62 (1979). [5] M. Kuranishi, On everywhere dense embedding of free groups in Lie groups,, Nagoya Math. J., 2 (1951), 63. [6] I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions,, Ergod. Theory Dynam. Systems, 23 (2003), 611. doi: 10.1017/S0143385702001554. [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. [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. [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. [10] I. Melbourne, V. Niţică and A. Török, Transitivity of Heisenberg group extensions of hyperbolic systems,, to appear in Ergod. Theory Dynam. Systems., (). [11] V. Niţică, Examples of topologically transitive skew-products,, Discrete Contin. Dynam. Systems, 6 (2000), 351. [12] V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms,, Ergod. Theory Dynam. Systems, 25 (2005), 257. doi: 10.1017/S0143385704000471. [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. [14] J. Schreier and S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe,, Fundamenta Math., 24 (1935), 302. [15] T.-S. Wu, The union of compact subgroups of an analytic group,, Trans. Amer. Math. Soc., 331 (1992), 869. doi: 10.2307/2154147.

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##### 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. [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. [3] M. Goto, A theorem on compact semi-simple groups,, J. Math. Soc. Japan, 1 (1949), 270. doi: 10.2969/jmsj/00130270. [4] M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups,, Graduate Texts in Mathematics, 62 (1979). [5] M. Kuranishi, On everywhere dense embedding of free groups in Lie groups,, Nagoya Math. J., 2 (1951), 63. [6] I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions,, Ergod. Theory Dynam. Systems, 23 (2003), 611. doi: 10.1017/S0143385702001554. [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. [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. [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. [10] I. Melbourne, V. Niţică and A. Török, Transitivity of Heisenberg group extensions of hyperbolic systems,, to appear in Ergod. Theory Dynam. Systems., (). [11] V. Niţică, Examples of topologically transitive skew-products,, Discrete Contin. Dynam. Systems, 6 (2000), 351. [12] V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms,, Ergod. Theory Dynam. Systems, 25 (2005), 257. doi: 10.1017/S0143385704000471. [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. [14] J. Schreier and S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe,, Fundamenta Math., 24 (1935), 302. [15] T.-S. Wu, The union of compact subgroups of an analytic group,, Trans. Amer. Math. Soc., 331 (1992), 869. doi: 10.2307/2154147.
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