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Abstract
We introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ≥ 0 such that for any pair of scales 0 < r < R , any ball of radius R may be covered by a constant times (R/r)^{α} balls of radius r. To each θ ∈ (0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/ log r=θ. The resulting ‘dimension spectrum’ (as a function of θ) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding ‘lower spectrum’, motivated by the lower dimension, which acts as a dual to the Assouad spectrum.
We conduct a detailed study of these dimension spectra; including analytic, geometric, and measureability properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with subexponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under biHölder maps for Assouad dimension and the provision of new biLipschitz invariants.
We conduct a detailed study of these dimension spectra; including analytic, geometric, and measureability properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with subexponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under biHölder maps for Assouad dimension and the provision of new biLipschitz invariants.
Original language  English 

Pages (fromto)  273328 
Number of pages  56 
Journal  Advances in Mathematics 
Volume  329 
Early online date  23 Feb 2018 
DOIs  
Publication status  Published  30 Apr 2018 
Keywords
 Assouad dimension
 Lower dimension
 Boxcounting dimension
 Continuity
 Measureability
 Unwinding spirals
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Dive into the research topics of 'New dimension spectra: finer information on scaling and homogeneity'. Together they form a unique fingerprint.Projects
 1 Finished

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
1/09/16 → 30/06/18
Project: Fellowship
Profiles

Jonathan Fraser
 School of Mathematics and Statistics  Director of Research
 Pure Mathematics  Professor
 Centre for Interdisciplinary Research in Computational Algebra
Person: Academic