December 2, 2009
Hanna Uscka-Wehlou (Uppsala): Some combinatorial problems related to digital straight lines with irrational slopes and to balanced aperiodic words
Abstract:
There is a very close relationship between digital straight lines and
binary words with the property of balance. During my Ph.D. studies I
described, by means of continued fractions, the run hierarchical
structure of digital straight lines (equivalently, of upper mechanical
words) with irrational positive slopes less than one. I defined two
equivalence relations on the set of slopes. Both relations are defined
on the continued fraction elements of the slopes and they have their
interpretation in the run hierarchical structure. This causes some
questions with combinatorial nature, for example, about the description
of the continued fraction expansion of slopes in each equivalence class.
Another combinatorial piece is the fixed point theorem for the lines and
words. The fixed points are the lines (words) with self-balanced
construction in terms of long and short runs on all digitization levels.