Given a substitution $1\to 100$, $0\to 01$, then we have $1\to 100\to1000101\to10001010110001100\to\cdots$, we denote this limits (fixed point of this substitution) as $(a_n)$, given $\beta>1 $. How can we find the value $\sum_{n=1}^{\infty}\dfrac{a_n}{\beta^n}$?
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$\begingroup$ Why do you need to know this? Some context would make it a better question. $\endgroup$– Anthony QuasCommented Oct 8, 2014 at 16:08
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1$\begingroup$ What sort of answer do you expect? Even for the Thue-Morse sequence, for instance, and $\beta=2$, this value is known to be transcendental (mathworld.wolfram.com/Thue-MorseConstant.html). $\endgroup$– Nikita SidorovCommented Oct 8, 2014 at 18:08
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$\begingroup$ Yes, I realize that the the value of $\sum_{n=1}^{\infty}a_n\beta^{-n}$ may not be calculated easily, so I can estimate this value. I think that we can find a subshift of finite type, and the problem then can be translated over. I remember the Mauldin-Williams' graph-directed self-similar sets could be one of the choices. $\endgroup$– Ben BenCommented Oct 9, 2014 at 7:33
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$\begingroup$ Subshift of finite type for what? $\endgroup$– Nikita SidorovCommented Oct 9, 2014 at 20:10
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$\begingroup$ The substitution is defined by that rule and the forbidden blocks can appear in the sequence(fixed points of substitution). We can define a SFT, of course, the SFT may be big. $\endgroup$– Ben BenCommented Oct 12, 2014 at 14:55
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