By Schaaf W.L.

ISBN-10: 0873531205

ISBN-13: 9780873531207

**Read Online or Download A bibliography of recreational mathematics (5 vols.) PDF**

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**Additional resources for A bibliography of recreational mathematics (5 vols.)**

**Sample text**

The set ω(f ) = g | there exists (nj )j ∈N with lim nj = ∞ such that lim S j →∞ j →∞ nj (f ) = g , is a periodic orbit of S. Proof. Let κ be the image-part-map associated with H and V and let Per κ be the set of periodic points of κ. , γfγ , R(f ) = γ ∈Per κ 34 2 Expanding endomorphisms and substitutions with the usual convention R(f )(γ ) = ∅ for γ ∈ Per κ. 1), We begin by proving lim n→∞ (S n (f ), S n (R(f ))) = 0. This means that the ω-limit set of f is determined by the ω-limit set of the restriction of f on Per κ.

There is a natural projection of p : (Z, A) → (N, A) deﬁned by p(f ) = f |N . There does not exist a natural embedding (N, A) into (Z, A); (N, A) into but there exists a natural embedding j of i(f )(x j ) = f (x j ) ∅ (Z, A) deﬁned by if j ≥ 0 if j < 0 The map H : Z → Z deﬁned by H (x j ) = x 2j is a monomorphism of Z and H (Z) ⊂ Z is a subgroup of index 2. Since Z is generated by x, there exists an induced norm on Z which we denote by and which is deﬁned by x j = |j |. t. the generating set {x}.

2. Consider H :L→L ⎛ ⎞ ⎛ ⎞ 1 x z 1 2x 4z ⎝ 0 1 y ⎠ → ⎝ 0 1 2y ⎠ . t. the norm r ). A (left)-residue set is given by V = {b 1 a 2 c 3 | 1 , 2 ∈ {0, 1} and 3 ∈ {0, 1, 2, 3}}. The residue set V is not a complete digit set. The ﬁxed points of κ are e, a −1 , b−1 , c−1 , b−1 c−1 , a −1 c−1 , b−1 a −1 c−1 . Moreover, these are the only periodic points of κ. The next theorem provides a simple criterion for the existence of complete digit sets. 7. t. the discrete norm with expansion ratio C > 2 and let H ( ) be of index d ∈ N.