Dimensions of invariant sets of expanding maps (Q1918064)

Let \(M\) be a \(C^\infty\)-Riemannian manifold and \(f\) a \(C^2\)-map of \(M\) to itself. Let \(\Lambda\) be a compact \(f\)-invariant subset of \(M\) such that \(f\) is expanding on \(\Lambda\). It is proved that the Hausdorff and lower and upper box dimensions of \(\Lambda\) belong to each of the intervals \([h/\sigma_1,h/ \sigma_0]\) and \([m+P/ \sigma_0,m+P/ \sigma_1]\), where \(h\) is the topological entropy of \(f_{|\Lambda}\), \(\sigma_0\) and \(\sigma_1\) are the smallest and largest uniform Lyapunov exponents, respectively, \(m\) is the dimension of \(M\) and \(P\) is the topological pressure of \(f_{|\Lambda}\) for the function \(-\log |Df(x) |\) (it is nonpositive). The result is deduced after proving a measure-theoretic analogue of it. It is well-known that many self-similar sets (fractals) can be realised as invariant sets of expanding maps as above, and the author notes that few effective ways are known to calculate the Hausdorff and box dimensions of sets invariant under nonlinear maps in higher dimensions.