Increase of entropy under convolution and selfsimilar sets with overlaps 

Author:  Pyörälä, Aleksi^{1} 
Organizations: 
^{1}University of Oulu, Faculty of Science, Mathematics 
Format:  ebook 
Version:  published version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.6 MB) 
Pages:  117 
Persistent link:  http://urn.fi/URN:NBN:fi:oulu201904251540 
Language:  English 
Published: 
Oulu : A. Pyörälä,
2019

Publish Date:  20190508 
Thesis type:  Master's thesis 
Tutor: 
Suomala, Ville Wu, Meng 
Reviewer: 
Suomala, Ville Wu, Meng 
Description: 
Abstract Sets that consist of finitely many smallerscale copies of itself are known as selfsimilar. Due to the likely irregularity in their structure, the size of these sets is often measured in the form of dimension. The existence of tools that can be used to calculate this quantity depends greatly on whether the cylinders of which the set consists of are sufficiently separated from each other. If this is the case, the dimension of the set is known to equal its similarity dimension, a quantity that is relatively easy to calculate. There is a longstanding open conjecture stating that, for a general set on the real line, the only case in which the dimension of the set does not equal its similarity dimension, is when at some scale there is an exact overlap among the cylinders of the set. The main result in this thesis is a step towards showing that this is indeed the case; in the presence of an exact overlap, the distance between the cylinders of the set decreases exponentially. This result is due to M. Hochman and it appeared in his paper “On selfsimilar sets with overlaps and inverse theorems for entropy” (2012) and forms the basis of our discussion in Section 4. In Section 1, we analyse the growth of entropy of a probability measure under convolution. The main result of this section is a generalization of the Freiman theorem from additive combinatorics to the fractal regime, stating that if the entropy of a convolution measure is not too large, then one of the marginal measures has to be either locally uniform or locally atomic. This result is also due to Hochman and is one of the main tools used in proving the results of Section 4. In Sections 2 and 3, we introduce the concepts of dimension and the main tools required in understanding the structure of a selfsimilar set or measure with sufficient separation conditions in place. Most of the results here can be found in any textbook concerning fractal geometry, e.g. Falconer’s “Fractal Geometry” (1990). see all

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Copyright information: 
© Aleksi Pyörälä, 2019. This publication is copyrighted. You may download, display and print it for your own personal use. Commercial use is prohibited. 