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Dimension and measure for generic continuous images

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Date
2013
Author
Balka, Richard
Farkas, Abel
Fraser, Jonathan M.
Hyde, James T.
Keywords
Hausdorff dimension
Packing dimension
Topological dimension
Baire category
Prevalence
Continuous functions
QA Mathematics
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Abstract
We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into R-n. The key question is 'what is the generic dimension of f(X)?' and we consider two different approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, n, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of n and the topological dimension of X. We also study, the typical Hausdorff and packing measures of f (X) and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of f (X) and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.
Citation
Balka , R , Farkas , A , Fraser , J M & Hyde , J T 2013 , ' Dimension and measure for generic continuous images ' , Annales Academiae Scientiarum Fennicae-Mathematica , vol. 38 , no. 1 , pp. 389-404 . https://doi.org/10.5186/aasfm.2013.3819
Publication
Annales Academiae Scientiarum Fennicae-Mathematica
Status
Peer reviewed
DOI
https://doi.org/10.5186/aasfm.2013.3819
ISSN
1239-629X
Type
Journal article
Rights
Copyright © 2013 by Academia Scientiarum Fennica. This is an Open Access journal.
Description
This work is supported by EPSRC Doctoral Training Grants
Collections
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/3902

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