Maximum Local Lyapunov Dimension Bounds the Box Dimension of Chaotic Attractors

Brian R. Hunt

Nonlinearity 9 (1996), 845-852.

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We prove a conjecture of Il'yashenko, that for a $C^1$ map in {\bf R}$^n$ which locally contracts $k$-dimensional volumes, the box dimension of any compact invariant set is less than $k$. This result was proved independently by Douady and Oesterl\'e and by Il'yashenko for Hausdorff dimension. An upper bound on the box dimension of an attractor is valuable because, unlike a bound on the Hausdorff dimension, it implies an upper bound on the dimension needed to embed the attractor. We also get the same bound for the fractional part of the box dimension as is obtained by Douady and Oesterl\'e for Hausdorff dimension. This upper bound can be characterized in terms of a local version of the Lyapunov dimension defined by Kaplan and Yorke.

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