Dimensions of attractors in pinched skew products

TL;DR

This paper investigates the dimensions of strange non-chaotic attractors in pinched skew products, showing that various dimensions are equal to one, confirming a longstanding conjecture and highlighting differences between Hausdorff and box-counting dimensions.

Contribution

It proves that Hausdorff, pointwise, and information dimensions of these attractors are all equal to one, resolving a conjecture from 1989.

Findings

Hausdorff, pointwise, and information dimensions are all equal to one

Physical measure is rectifiable, implying pointwise dimension equals one

Box-counting dimension remains two, highlighting dimension discrepancies

Abstract

We study dimensions of strange non-chaotic attractors and their associated physical measures in so-called pinched skew products, introduced by Grebogi and his coworkers in 1984. Our main results are that the Hausdorff dimension, the pointwise dimension and the information dimension are all equal to one, although the box-counting dimension is known to be two. The assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable. Our findings confirm a conjecture by Ding, Grebogi and Ott from 1989.