The Hausdorff dimension of self-similar sets under a pinching condition

Gu, Xiao Ping

doi:10.1090/S0002-9939-1993-1181166-1

The Hausdorff dimension of self-similar sets under a pinching condition
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Proc. Amer. Math. Soc. 118 (1993), 1281-1289 Request permission

Abstract:

We study self-similar sets in the case where the construction diffeomorphisms are not necessarily conformal. Using topological pressure we give an upper estimate of the Hausdorff dimension, when the construction diffeomorphisms are ${C^{1 + \kappa }}$ and satisfy a $\kappa$-pinching condition for some $\kappa \leqslant 1$. Moreover, if the construction diffeomorphisms also satisfy the disjoint open set condition we then give a lower bound for the Hausdorff dimension.

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