Combinatorical and Algebraic Structures Seminar

Date:	29.9.2015
Speaker:	Artur Siemaszko, WMiI, Uniwersytet Warmińsko-Mazurski
Title:	Counting Adler-Weiss partitions of toral automorphisms via Sturmian words and substitution tilings
Abstract:	For an automorphism of two the dimensional torus by a \emph{Adler-Weiss partition} we name a Markov partition into two rectangles (or parallelograms depending on coordinates). It is shown that there are finitely many Adler-Weiss partitions. Moreover coding of the chosen eigendirection of the automorphism by any Berg partition gives the same Sturmian sequence. Those Sturmian sequences exhibit existence of pattern structure of palindromes (which is relatively easy to examine geometrically and quite difficult to prove in terms of combinatorics). Using this structure we are able to show that there exactly $\mathrm{ENT}\left(\frac{a+b+c+d}{2}\right)$ Adler-Weiss partitions with a transition matrix $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$. This formula is related to the fact, due to P.~S\'{e}\'{e}bold, that there are exactly $a+b+c+d-1$ substitutions fixing the Sturmian sequence under consideration. Moreover there are $2(p_1+p_2+\cdots+p_n)$ different transition matrices where $ (p_1,p_2,\ldots,p_n)$ is a period of the continued fraction expansion of a slope of the principal eigendirection.

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