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Journal of the London Mathematical Society Advance Access originally published online on February 17, 2009
Journal of the London Mathematical Society 2009 79(2):445-464; doi:10.1112/jlms/jdn070
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© 2009 London Mathematical Society

Which beta-shifts have a largest invariant measure?

Vasso Anagnostopoulou and Oliver Jenkinson

School of Mathematical Sciences
Queen Mary, University of London
Mile End Road
London
E1 4NS
United Kingdom
vaa@maths.qmul.ac.uk
www.maths.qmul.ac.uk~omj

For a given beta-shift, the lexicographic order induces a partial order (known as first-order stochastic dominance) on the collection of its shift-invariant probability measures. We characterize those beta-shifts for which this partial order has a largest element. These beta-shifts are all of finite type, and their lexicographically largest point is a periodic sequence of a particular kind: it is Sturmian (that is, its shift-orbit is combinatorially equivalent to a rotation) with weight-per-symbol either an integer, or equal to p/(ap + 1) for some a, p ≥ 1, or equal to A + p/(p + 1) for some p ≥ 1 and A ≥ 2. In these cases, the largest invariant measure is precisely the unique one supported by the shift-orbit of the lexicographically largest point in the beta-shift.

2000 Mathematics Subject Classification 37B10 (primary), 11A63, 37A05, 37A45, 37D20, 37E05, 37E15, 37E45 (secondary).

The first author was partially supported by an EPSRC PhD Studentship and an Erwin Schrödinger Institute Junior Research Fellowship. The second author was partially supported by an EPSRC Advanced Research Fellowship.

Received May 20, 2008; published online February 17, 2009.


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