120 research outputs found
Outer billiard outside regular polygons
We consider outer billiard outside regular convex polygons. We deal with the
case of regular polygons with sides, and we describe the
symbolic dynamics of the map and compute the complexity of the language.Comment: 53 pages, 12 figure
Almost everywhere balanced sequences of complexity
We study ternary sequences associated with a multidimensional continued
fraction algorithm introduced by the first author. The algorithm is defined by
two matrices and we show that it is measurably isomorphic to the shift on the
set of directive sequences. For a given set
of two substitutions, we show that there exists a -adic sequence
for every vector of letter frequencies or, equivalently, for every directive
sequence. We show that their factor complexity is at most and is
if and only if the letter frequencies are rationally independent if and only if
the -adic representation is primitive. It turns out that in this
case, the sequences are dendric. We also prove that -almost every
-adic sequence is balanced, where is any shift-invariant
ergodic Borel probability measure on giving a positive
measure to the cylinder . We also prove that the second Lyapunov
exponent of the matrix cocycle associated with the measure is negative.Comment: 42 pages, 9 figures. Extended and augmented version of
arXiv:1707.0274
Diophantine properties of real numbers generated by finite automata
We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the -adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit
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