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Mathematics Lecture : Generalized shifts of finite type associated to an integer matrix

Dr. Sharvari Tikekar,TIFR Mumbai
Dr. Sharvari Tikekar,TIFR Mumbai
When Sep 22, 2022
from 04:00 PM to 05:00 PM
Where Auditorium, Ground Floor, TIFR CAM
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Shifts of finite type are one of the fundamental objects in the field of symbolic dynamics. These are formed as the spaces of one-sided or two-sided sequences over finite set of symbols where certain finitely many words are forbidden. They exhibit a natural association with a 0 − 1 matrix. In this talk we will establish some necessary preliminaries related to the one-sided shifts of finite type associated to any non-negative integer matrix. Words which correspond to the matrix entries greater than 1 are thought to have multiplicity and thus called repeated words. Now, for any given collection F of forbidden words and R of repeated words, we define two notions: multiplicity of a word and generalized language. We study the shift determined by F and R, and obtain necessary and sufficient conditions for when the language of this shift is precisely the generalized language. We

compute the topological entropy of this shift and discuss some properties of Markov measure on the shift. We obtain an expression for the generating function that enumerates the number of allowed words of fixed length in this shift. When R is empty, the theory of generating functions is studied by Guibas and Odlyzko. These results provide an expression for the Perron root and eigenvectors of any irreducible non-negative integer matrix. The theory also has applications in obtaining an alternate
definition of Parry measure on the associated shift. When R is empty, these results are studied by Agarwal N. (IISER B) and Haritha C.(TIFR). This is a joint work with Agarwal N. and Haritha C.

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