MAXIMUM LOCAL LYAPUNOV DIMENSION BOUNDS THE BOX DIMENSION OF CHAOTIC ATTRACTORS

We prove a conjecture of Il'yashenko, that for a C-1 map in R(n) which locally contracts k-dimensional volumes, the box dimension of any com pact invariant set is less than k. This result was proved independentl y by Douady and Oesterle and by Il'yashenko for Hausdorff dimension. A n 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 D ouady and Oesterle for Hausdorff dimension. This upper bound can be ch aracterized in terms of a local version of the Lyapunov dimension defi ned by Kaplan and Yorke.