Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

• We consider the distribution of the orbits of the number 1 under the -transformations as varies. Mainly, the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set is determined, where is a given point in and is a sequence of integers tending to infinity as . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of with a common prefix in the expansion of 1) in the parameter space .

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

beta-expansion

Diophantine approximation

Hausdorff dimension

Publication and Content Type

art (subject category)

ref (subject category)