Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 945-960

SOME OBSERVATIONS ON KÄENMÄKI MEASURES

Ian D. Morris

University of Surrey, Department of Mathematics
Guildford GU2 7XH, United Kingdom; i.morris 'at' surrey.ac.uk

Abstract. In this note we investigate some properties of equilibrium states of affine iterated function systems, sometimes known as Käenmäki measures. We give a simple sufficient condition for Käenmäki measures to have a gap between certain specific pairs of Lyapunov exponents, partially answering a question of Bárány, Käenmäki and Koivusalo. We also give sharp bounds for the number of ergodic Käenmäki measures in dimensions up to 4, answering a question of Bochi and the author within this range of dimensions. Finally, we pose an open problem on the Hausdorff dimension of self-affine measures which may be reduced to a statement concerning semigroups of matrices in which a particular weighted product of absolute eigenvalues is constant.

2010 Mathematics Subject Classification: Primary 28A80, 37D35, 37H15.

Key words:

Reference to this article: I. D. Morris: Some observations on Käenmäki measures. Ann. Acad. Sci. Fenn. Math. 43 (2018), 945-960.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4357