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Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms

Published online by Cambridge University Press:  19 September 2008

Lorenzo J. Diaz  and
Marcelo Viana
Lorenzo J. Diaz
Affiliation:
IMPA, Estrada Dona Castorina 110, 22.460-Rio de Janeiro-RJ, Brasil
Marcelo Viana
Affiliation:
Departamento de Matemática, Fac. Ciências do Porto, 4000 Porto-Portugal
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Abstract

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We consider one-parameter families of torus diffeomorphisms that bifurcate from global hyperbolic maps (Anosov) to DA maps (derived from Anosov). For an open set of these families, we show that the Hausdorff dimension and limit capacity of the nonwandering set are not continuous across the bifurcation. We also study the behaviour of equilibrium measures near the bifurcation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 3 , September 1989 , pp. 403 - 425
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Bowen, R.. Equilibrium states and ergodic theory of Axiom A diffeomorphisms. Lecture Notes in Mathematics 470. Springer Verlag: New York, 1975.Google Scholar
[2]Bowen, R.. On Axiom A diffeomorphisms. Conference Board Math. Sciences 35 Amer. Math. Soc. (1977).Google Scholar
[3]Hirsch, M. & Pugh, C.. Stable manifolds and hyperbolic sets. Global Analysis. Proc. Symp. in Pure Math. XIV, Amer. Math. Soc. (1970), 133–163.Google Scholar
[4]McCluskey, H. & Manning, A.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1983), 231–260.CrossRefGoogle Scholar
[5]Manning, A.. A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 451–459.CrossRefGoogle Scholar
[6]Manning, A.. Errata to ‘Hausdorff dimension for horseshoes’. Ergod. Th. & Dynam. Sys. 5 (1985), 319.CrossRefGoogle Scholar
[7]Marstrand, J.M.. The dimension of cartesian products of sets. Proc. Cambridge Phil. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
[8]Palis, J. & Viana, M.. On the continuity of Hausdorff dimensions and limit capacity for horseshoes. Topics in Dynamics, Proceedings Chilean Symp., Lecture Notes in Math. 1331. Springer Verlag: New York, 1988.CrossRefGoogle Scholar
[9]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Bras, de Mat. 9 (1978), 83–87.CrossRefGoogle Scholar
[10]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[11]Takens, F.. Limit capacity and Hausdorff dimension of dynamically defined Cantor sets. Topics in Dynamics, Proceedings Chilean Symp., Lecture Notes in Math. 1331. Springer Verlag: New York, 1988.Google Scholar
[12]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1976), 937–971.CrossRefGoogle Scholar
[13]Williams, R.. The ‘DA’ maps of Smale and structural stability. Global Analysis, Proc. Symp. in Pure Math. XIV Amer. Math. Soc. (1970), 329–334.Google Scholar