Dimension and measure for generic continuous images

Richard Balka*, Abel Farkas, Jonathan M. Fraser, James T. Hyde

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)
2 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)389-404
Number of pages16
JournalAnnales Academiae Scientiarum Fennicae-Mathematica
Volume38
Issue number1
DOIs
Publication statusPublished - 2013

Keywords

  • Hausdorff dimension
  • Packing dimension
  • Topological dimension
  • Baire category
  • Prevalence
  • Continuous functions

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