Abstract
AbstractDimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in understanding how these different notions fit together, as well as how their subtle differences give rise to different behaviour. Here we survey a new approach in dimension theory, which seeks to unify the study of individual dimensions by viewing them as different facets of the same object. For example, given two notions of dimension, one may be able to define a continuously parameterised family of dimensions which interpolates between them. An understanding of this ‘interpolation function’ therefore contains more information about a given object than the two dimensions considered in isolation. We pay particular attention to two concrete examples of this, namely the Assouad spectrum, which interpolates between the box and (quasi-)Assouad dimension, and the intermediate dimensions, which interpolate between the Hausdorff and box dimensions.KeywordsDimension theoryHausdorff dimensionBox dimensionAssouad dimensionAssouad spectrumIntermediate dimensionsMathematics Subject Classifications (2010)Primary: 28A80Secondary: 37C45
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
