Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 389-404

DIMENSION AND MEASURE FOR GENERIC CONTINUOUS IMAGES

Richárd Balka, Ábel Farkas, Jonathan M. Fraser and James T. Hyde

Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics
P.O. Box 127, H-1364 Budapest, Hungary; balka.richard 'at' renyi.mta.hu

University of St Andrews, Mathematical Institute
North Haugh, St Andrews, Fife, KY16 9SS, Scotland; af56 'at' st-andrews.ac.uk

University of St Andrews, Mathematical Institute
North Haugh, St Andrews, Fife, KY16 9SS, Scotland; jmf32 'at' st-andrews.ac.uk

University of St Andrews, Mathematical Institute
North Haugh, St Andrews, Fife, KY16 9SS, Scotland; jth4 'at' st-andrews.ac.uk

Abstract. We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into Rⁿ. 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.

2010 Mathematics Subject Classification: Primary 28A80, 28A78, 54E52, 54C05.

Key words: Hausdorff dimension, packing dimension, topological dimension, Baire category, prevalence, continuous functions.

Reference to this article: R. Balka, Á. Farkas, J.M. Fraser and J.T. Hyde: Dimension and measure for generic continuous images. Ann. Acad. Sci. Fenn. Math. 38 (2013), 389-404.

Full document as PDF file

doi:10.5186/aasfm.2013.3819