On multifractal analysis and local dimensions of measures
1University of Oulu, Faculty of Science, Mathematics
|Online Access:||PDF Full Text (PDF, 0.6 MB)|
|Persistent link:|| http://urn.fi/URN:NBN:fi:oulu-202107108795
Oulu : R. Anttila,
|Publish Date:|| 2021-07-12
|Thesis type:||Master's thesis
This thesis consists of two independent parts. Part I serves as an introduction to multifractal analysis and establishes key concepts in the field, whereas Part II considers a generalisation of multifractal analysis in doubling metric spaces.
In multifractal analysis, one is interested in finely analysing the dimensional properties of measures by considering the behaviour of different level sets of the local dimension map. For some measures, these level sets are non-empty and exhibit fractal scaling according to a continuous spectrum of dimensions. These measures are called multifractal measures, and they are the main object of study in this thesis. The continuous spectrum of dimensions is called the multifractal spectrum of the measure, and it contains detailed information about the dimensional properties of the measure. For example, in the case of self-similar measures, the Hausdorff dimension of the underlying iterated function system (the support of the measure) and the Hausdorff dimension of the measure itself are easily obtained from the multifractal spectrum.
A celebrated principle in multifractal analysis is the multifractal formalism. With origins in physics literature, more specifically the study of turbulence in fluids, this heuristic principle states that the multifractal spectrum of a measure can be obtained from the Lq-spectrum of the measure via a simple Legendre transform. Usually the aim in the theory of multifractals is to obtain a rigorous notion of the multifractal formalism, and this has already been done for a wide variety of measures. In Part I of the thesis, we refrain to perhaps the most simple non-trivial case, namely the class of strongly separated self-similar measures. Our goal by the end of Part I is to establish the multifractal formalism rigorously for these measures and along the way, we introduce some basic fractal geometry, and develop the theory of multifractals. The main references of Part I are the two excellent textbooks by Falconer, “Fractal Geometry” (1990) and “Techniques in Fractal Geometry” (1997).
The second part of the thesis arises from the authors work as a university trainee in the Fractal Geometry Research Group at the Department of Mathematics in the summer of 2020. The discussion revolves mainly around local variants of Lq- and entropy dimensions of measures in doubling metric spaces, and aim of this part is to correct a slight inaccuracy found in an article by Käenmäki, Rajala and Suomala. To do this, we define restricted Lq-dimensions and show that they can be calculated by considering partitions of the underlying space. This simplifies the estimation of the local variants as well and as our main result, we obtain a correct proof for relationships between the local entropy dimensions and Lq-dimensions for (almost) arbitrary measures in doubling metric spaces.
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