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Fractal dimension estimate for invariant set in complete Riemannian manifold

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  • Qu, Chengqin
  • Zhou, Zuoling

Abstract

In this paper, we give a simple upper bounds for the fractal dimensions of a forward invariant set and a negatively invariant set of a C1-diffeomorphism of the complete Riemannian manifold with non-negative Ricci curvature. These results generalize the corresponding result of C. Robinson [Dynamics systems: stability symbolic dynamics and chaos. 2nd ed. Boca Raton, London, New York, Washington: CRC Press; 1999] in Rn, and weakens the condition to the singular values of the tangent map in [Boichenko VA, Franz A, Leonov GA, Reitmann V. Hausdorff and fractal dimension estimates for invariant sets of non-injective maps. Z Anal Anw 1998;17:207–23]. Finally, as application, we give the upper bound of the fractal dimension of an invariant set of the flow on Riemannian manifold.

Suggested Citation

  • Qu, Chengqin & Zhou, Zuoling, 2007. "Fractal dimension estimate for invariant set in complete Riemannian manifold," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1165-1172.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:5:p:1165-1172
    DOI: 10.1016/j.chaos.2005.08.207
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