By Kenneth B Stolarsky

ISBN-10: 0824761022

ISBN-13: 9780824761028

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**Example text**

N and at least one ui is . 5. If f ∈ F(A) is proper, deﬁne f∗= ∞ f n and f + = n=0 ∞ f n (a pointwise ﬁnite sum), n=1 with f 0 = 1 = 1 · = . 6. The rational operations in F(A) are sum (+), product (·), multiplication by real numbers (tw), and ∗ : f → f ∗ . The family of rational series consists of those f ∈ F(A) that can be obtained by starting with a ﬁnite set of polynomials in F(A) and applying a ﬁnite number of rational operations. 7. A language L ⊂ A∗ is rational if and only if its characteristic series F(w) = 1 if w ∈ L, 0 if w ∈ /L (54) is rational.

F is a member of a stable ﬁnitely generated submodule of FR+ (A). F is rational. The measure ν is the image under a one-block map of a shift-invariant one-step Markov probability measure µ. In the latter case, the measure ν is ergodic if and only if it is possible to choose µ ergodic. In the next few sections we sketch the proof of this theorem. 2 Proof that a series is linearly representable if and only if it is a member of a stable ﬁnitely generated submodule of F (A) Suppose that F is linearly representable by (x, φ, y).

1) and such that there is a unique relatively maximal measure µ above any fully supported Markov measure ν, but the measure µ is not Markov, and it is not even soﬁc. We use vertex shifts of ﬁnite type. The alphabet for the domain subshift is {a1 , a2 , b} (in that order for indexing purposes), and the factor map (onto the 2-shift ( 2 , σ )) is the one-block code π which erases subscripts. The transition diagram and matrix A for the domain shift of ﬁnite type ( A , σ ) are a1 b ( ) 1 1 0 1 1 1 1 1 .

### Algebraic numbers and Diophantine approximation by Kenneth B Stolarsky

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