# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 99-105. 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), 517-551. doi: 10.1017/S0143385704000355. [3] M. Goto, A theorem on compact semi-simple groups, J. Math. Soc. Japan, 1 (1949), 270-272. doi: 10.2969/jmsj/00130270. [4] M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups, Graduate Texts in Mathematics, Vol. 62, Springer-Verlag, New York, 1979. (translated from the second Russian edition by Robert G. Burns) [5] M. Kuranishi, On everywhere dense embedding of free groups in Lie groups, Nagoya Math. J., 2 (1951), 63-71. [6] I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions, Ergod. Theory Dynam. Systems, 23 (2003), 611-619. 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), 725-746. 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), 355-363. [9] I. Melbourne, V. Niţică and A. Török, Transitivity of Euclidean-type extensions of hyperbolic systems, Ergod. Theory Dynam. Systems, 29 (2009), 1585-1602. 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), 351-360. [12] V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms, Ergod. Theory Dynam. Systems, 25 (2005), 257-269. 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), 259-278. 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), 302-304. [15] T.-S. Wu, The union of compact subgroups of an analytic group, Trans. Amer. Math. Soc., 331 (1992), 869-879. 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), 99-105. 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), 517-551. doi: 10.1017/S0143385704000355. [3] M. Goto, A theorem on compact semi-simple groups, J. Math. Soc. Japan, 1 (1949), 270-272. doi: 10.2969/jmsj/00130270. [4] M. I. Kargapolov and J. I. Merzljakov, Fundamentals of the theory of groups, Graduate Texts in Mathematics, Vol. 62, Springer-Verlag, New York, 1979. (translated from the second Russian edition by Robert G. Burns) [5] M. Kuranishi, On everywhere dense embedding of free groups in Lie groups, Nagoya Math. J., 2 (1951), 63-71. [6] I. Melbourne and M. Nicol, Stable transitivity of Euclidean group extensions, Ergod. Theory Dynam. Systems, 23 (2003), 611-619. 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), 725-746. 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), 355-363. [9] I. Melbourne, V. Niţică and A. Török, Transitivity of Euclidean-type extensions of hyperbolic systems, Ergod. Theory Dynam. Systems, 29 (2009), 1585-1602. 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), 351-360. [12] V. Niţică and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms, Ergod. Theory Dynam. Systems, 25 (2005), 257-269. 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), 259-278. 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), 302-304. [15] T.-S. Wu, The union of compact subgroups of an analytic group, Trans. Amer. Math. Soc., 331 (1992), 869-879. doi: 10.2307/2154147.
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