Abstract
We describe some recent analytic results on the co-existence of symmetry and chaotic dynamics in equivariant dynamics. We emphasize the case of skew-products and stably SRB attractors
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References
R Adler, B Kitchens and M Shub. `Stably ergodic skew products’Discrete and Continuous Dynamical Systems2 (1996), 349–456 (Errata in 5 (1999)).
D V Anosov. `Geodesic flows on closed Riemann manifolds with negative curvature’Proc. of the Steklov Inst. of Math., 90 (1967).
P. Ashwin and I. Melbourne. Symmetry groups of attractors.Arch. Rat. Mech. Anal.126 (1994) 59–78.
R Bowen.Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms.Springer Lect. Notes in Math. 470, 1975.
M I Brin. `Topology of group extensions of Anosov systems’Mathematical Notes of the Acad. of Sciences of the USSR18(3) (1975), 858–864.
M I Brin. `Topological transitivity of one class of dynamic systems and flows of frames on manifolds of negative curvature’Funkts. Anal. Prilozh9, No. 1, (1975), 9–19.
M I Brin and J B Pesin. `Partially hyperbolic dynamical systems’Math. USSR Izvestia8, (1974), 177–218.
K Burns and A Wilkinson. `Stable ergodicity of skew products’ Ann.Sci. de l’Ecole Norm. Sup.32 (1999), 859–889.
K Burns, C Pugh, M Shub and A Wilkinson. `Recent results about stable ergodicity’, to appear inProc. AMS Summer Research Institute, Seattle, 1999.
P Chossat and M GolubitskySymmetry increasing bifurcations of chaotic attractorsPhysica D 32 (1988), 423–436.
U-R Fiebig. `Periodic points and finite group actions on shifts of finite types’Erg. Thy. l Dynam. Sys.13 (1993), 485–514.
M J Field. `Isotopy and stability of equivariant diffeomorphisms’Proc. London Math. Soc.(3) 46 (1983), 487–516.
M J Field. `Generating sets for compact semisimple Lie groups’Proc. Amer. Math. Soc.(1999), 3361–3365.
M J Field, I Melbourne and M Nicol. ‘Symmetric Attractors for Diffeomorphisms and Flows’Proc. London Math. Soc.72(3) (1996), 657–696.
M J Field and M Nicol. `Ergodic theory of equivariant diffeomorphisms; Markov partitions and Stable Ergodicity’, preprint.
M J Field and V Nitic¨¢. `Stable topological transitivity of skew and principal extensions’Nonlinearity14 (2001), 1–15.
M J Field and W Parry. `Stable ergodicity of skew extensions by compact Lie groups’Topology38, (1999), 167–187.
M Grayson, C Pugh and M Shub. `Stably ergodic diffeomorphisms’Annal of Math.140 (1994), 295–329.
A Katok and B Hasselblatt.Introduction to the Modern Theory of Dynamical Systems(Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.)
A Katok and A Kononenko. `Cocycle stability for partially hyperbolic systems’Math. Res. Lett.3 (1996), 191–210.
F Ledrappier and L S Young. `The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin’s entropy formula’Ann. of Math122 (1985), 509–539.
A N Livsic. `Cohomology of Dynamical Systems’Mathematics of the USSR Izvestija6(6), (1972), 1278–1301.
W Parry. `Skew-products of shifts with a compact Lie group’J. London Math. Soc.56(2) (1997), 400–404.
W Parry and M Pollicott. `Stability of mixing for toral extensions of hyperbolic systems’Tr. Mat. Inst. Steklova216 (1997), Din Sist. i Smezhnye Vopr., 354–363.
C Pugh and M Shub. `Stably ergodic dynamical systems and partial hyperbolicity’J. of Complexity13 (1997), 125–179.
D Ruelle and D Sullivan. `Currents, flows and diffeomorphisms’Topology14, (1975), 319–327
M Shub and A Wilkinson. `Pathological foliations and removable zero exponents’Invent. Math..139 (2000), 495–508.
L S Young. `Ergodic theory of chaotic dynamical systems’, Lect. Notes in Math. (FromTopology to Computation: Proceedings of the Smalefest(eds Hirsch, Marsden, Shub), Springer-Verlag, New York, Heidelberg, Berlin, 1993.)
P Walters. AnIntroduction to Ergodic Theory.(Springer Verlag, 1982).
A Wilkinson. ‘Stable ergodicity of the time-one map of a geodesic flow’Erg. Th. Dyn. Syst.18(6) (1998), 1545–1588.
R F Williams. One-dimensional non-wandering setsTopology6 (1967), 473–487.
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Field, M. (2003). Persistent Ergodicity and Stably Ergodic SRB Attractors in Equivariant Dynamics. In: Buescu, J., Castro, S.B.S.D., da Silva Dias, A.P., Labouriau, I.S. (eds) Bifurcation, Symmetry and Patterns. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7982-8_4
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DOI: https://doi.org/10.1007/978-3-0348-7982-8_4
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