The box dimension of invariant graphs over hyperbolic systems

Abstract

Let f be a C 1þ hyperbolic diffeomorphism on a basic set Ã. Suppose that g : à ! DiffðRÞ is such that g(x) uniformly contracts for each x 2 Ã. It is well known that there exists an invariant graph À for the skew product map Fðx, yÞ ¼ ð fðxÞ, gðxÞyÞ on à  R. When F is partially hyperbolic, we discuss the box dimension of À in terms of solutions to Bowen's pressure equation.

Hyperbolic graphs: Critical regularity and box dimension

“…Specifically, compare Section 3.2. More precisely, one of the main problems in [45] is that the Figure 1. Comparison between the estimates D 2 (γ) (straight line) and D 1 (γ) (convex curve) for the case d = dim B (Ξ) < 1.…”

“…A more general result that includes Theorem A has been announced in [45]. However, due to a serious flaw in the argument given in that paper, a complete proof for the statement in [45] does not exist so far. We will discuss this issue in detail in Section 1.5 below.…”

“…We note that the particular case of skew product systems with affine fiber maps and linear torus automorphisms in the base (see Section 3.1) is already covered by the results of [22] using Fourier analysis. A more general result that includes Theorem A has been announced in [45]. However, due to a serious flaw in the argument given in that paper, a complete proof for the statement in [45] does not exist so far.…”

“…An explicit example for this will be discussed in Section 3.2 (see Remark 3.4). This also points out an error in [45], whose main result can be seen to be false exactly because it asserts that the box dimension equals D 2 (γ) (and its generalized version corresponding to (1.11)) in situations where D 1 (γ) < D 2 (γ). Specifically, compare Section 3.2.…”

“…Specifically, compare Section 3.2. More precisely, one of the main problems in [45] is that the Intermediate Value Theorem (IVT) is applied to continuous functions defined only on Cantor sets. Even for situations where the formula for the box dimension in [45] is expected to be the correct one, we do not see how to fix this gap in a direct way.…”