Abstract
AbstractWe derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.
Highlights
We note that for a given θ it is not necessarily true that the Assouad spectrum is given by the expression after the limit in the definition of the quasi-Assouad dimension: this notion is by definition monotonic in θ, but the spectrum is not necessarily monotonic [6, §8]
It has recently been shown in [4] that dimqA F = limθ→1 dimθA F and, combining this with (1.1), we see that the Assouad spectrum necessarily interpolates between the upper box dimension and the quasi-Assouad dimension
The Assouad spectrum was computed by Fraser and Yu [7], and these results demonstrated that the quasi-Assouad and Assouad dimensions coincide by virtue of the spectrum reaching the Assouad dimension
Summary
3. Results Our main result is an explicit formula which gives the Assouad spectrum of our random self-affine sets almost surely. Log ni Corollary 3.2 demonstrates the striking difference between the Assouad and quasi-Assouad dimensions in the random setting.
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