Dimension estimates for iterated function systems and repellers. Part II

This is the second part of our study on the dimension theory of C¹ iterated function systems (IFSs) and repellers on Rᵈ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for C¹ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every C¹ IFS on Rᵈ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of C¹ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of C¹ IFSs on Rᵈ .