Dimension estimates for iterated function systems and repellers. Part II

Abstract

This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ 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 $C^1$ IFS on ${\Bbb R}^d$ 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 $C^1$ 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 $C^1$ IFSs on ${\Bbb R}^d$ .

Dimension estimates for iterated function systems and repellers. Part I

“…In the continuation of this paper [22] we verify that upper bound estimates of the dimensions of the attractors and ergodic measures in the previous paragraph give the exact values of the dimensions for some families of C 1 non-conformal IFSs in R d at least typically. 'Typically' means that the assertions hold for almost all translations of the system.…”

Exact dimension of Furstenberg measures

“…An open problem is to find conditions under which we have equality in (1.5). This is true when d = 2 and in some examples in higher dimensions, see [FS22] for the latest results and a history of similar problems, in particular for IFS. Define the function D µ by replacing κ(µ, ν) by the random walk entropy h RW (µ) := inf n 1 n H(µ * n ).…”

Notes on the Transversality Method for Iterated Function Systems—A Survey

“…Much later, this was sharpened and extended to a more general class of self-affine systems by T. Jordan, M. Pollicott, and K. Simon [22] in the framework of "self-affine transversality". Very recently, self-affine transversality was further extended to non-linear non-conformal systems by De-Jun Feng and Károly Simon [23].…”

“…(33)Denotec 1 = (2C 1 ) −β/α . Then {(ω,τ): ρ(ω,τ)≥(|y−z|/2C 1 ) 1/α } ρ(ω, τ) −β dµ(ω) dµ(τ) = c 1 |y−z| −β/α µ × µ) (ω, τ) : ρ(ω, τ) ≤ t −1/β dt ≤ c 1 |y−z| −β/α −γ µ /β dt |y − z| − β−γµ α (recall that γ µ < β),where we used(23). Finally,R 2d |y − z| − β−γµ α dη(y) dη(z) < ∞,in view of(26), and so I 2 r d by(33), as desired.…”