Möbius number systems with sofic subshifts

Transcription

1 Möbius number systems with sofic subshifts Petr Kůrka Center for Theoretical Study, Academy of Sciences and Charles University in Prague, Jilská, CZ Praha, Czechia. Abstract. A real Möbius iterative system is an action of a free semigroup of finite words acting via Möbius transformations on the extended real line. Its convergence space consists of all infinite words, such that the images of the Cauchy measure by the prefixes of the word converge to a point measure. A Möbius number system consists of a Möbius iterative system and a sofic subshift included in the convergence space, such that any point measure can be obtained as the limit of some word of the subshift. We give some sufficient conditions on sofic subshifts to form Möbius number systems. We apply our results to several systems based on continued fractions.. Introduction Classical number systems for compact real intervals such as decadic, binary, or binary signed number systems, can be obtained as naming functions of contractive iterative systems (see e.g., Barnsley [] or Edgar [3]). An iterative system (F a : X X) a A consists of continuous selfmaps of a compact metric space X indexed by a finite alphabet A. In contractive iterative systems, each infinite word u A N determines a unique point x = Φ(u) which is contained in all images F u F u F un (X) of the state space X by the prefixes of the word u. The range of the naming map Φ is a compact subset of X called the attractor of the system. The method works, however, only for compact subspaces of the real line or complex plane. It is therefore natural to look for number systems on a compactification of the real line such as the extended real line R which is homeomorphic to the unit circle T. A natural choice for the mappings are Möbius transformations, since these are the only holomorphic isomorphisms of the extended complex sphere C = C { }. However, since Möbius transformations are not contractions and are surjective, the Barnsley theorem does not work for them. In Kůrka [8] we have used convergence of measures (instead of convergence of sets), to obtain symbolic representations of the extended real line from iterative Möbius systems. The uniform measure on T is transferred via stereographic projection to the Cauchy measure on R. We say that an infinitte word u A N converges to a number x R, if the images of the Cauchy measure by the prefixes of the word u converge to the point measure δ x. In this case we say that u is a name of x and write Φ(u) = x. The domain of Φ is the convergence space X F A N. To get a number system, the range of Φ should be all R. Since Φ is usually not continuous on X F we look for a subshift Σ X F such that Φ(Σ) = R and Φ is continuous on Σ. In this case we say that (F, Σ) is a Möbius number system.

2 In [8] we have constructed several Möbius number systems using contractions to vertices of a regular polygon inscribed to the unit circle. In [7] we have obtained some results on topological dynamics of Möbius iterative systems. In the present paper we develop a theory of Möbius number systems with sofic subshifts. In Theorem 8 we characterize those Möbius iterative systems whose range is whole R. In the Convergence Theorem we give a sufficient condition on a sofic subshift Σ to satisfy Σ X F. In the Surjectivity Theorem 3 we give a sufficient condition for Φ(Σ) = R. Finally we show that for Möbius number systems whose coefficients are in a computable field (such as rational or quadratic numbers), efficient arithmetic online algorithms exist. We apply the theory to several systems based on continued fractions.. Möbius transformations A real, orientationpreserving Möbius transformation is a selfmap of the extended real line R := R { } of the form M (a,b,c,d) (x) = (ax + b)/(cx + d), where ad bc >. Möbius transformations act not only on R but also on the upper halfplane U := {z C : I(z) > }, where they preserve the hyperbolic metric ds = dz/i(z). The upper halfplane U is conformally isomorphic to the unit disc D = {z C : z < } via isomorphisms d : U D and u : D U given by d(z) = (z i)/(z + i), u(z) = i(z )/(z + ). On D we have disc Möbius transformations M (a,b,c,d) = d M (a,b,c,d) u which preserve the hyperbolic metric ds = dz/( z ). The transformation d maps i to and R to D = {z C : z = }, which we parametrize by T := R/πZ = [ π, π). We have mutually inverse transformations x : T R and t : R T given by x(t) = u(e it ) = tan t, t(x) = arg d(x) = arctanx, and circle Möbius transformations M (a,b,c,d) = t M (a,b,c,d) x : T T. Alternatively, we can view them as increasing continuous functions M : R R with M(t + π) = M(t)+ π. The distance on T yields the circle distance d(x, y) := min{ t(x) t(y), π t(x) t(y) } on R. Given a, b R, define the interval [a, b] and its circle length [a, b] by a < b [a, b] := {x R : a x b}, [a, b] = t(b) t(a), a b [a, b] := {x R : a x or x b}, [a, b] = π t(b) + t(a). Intervals [a, ] and [, b] are defined analogously. Define the circle derivation, the square of the trace, and the norm of a Möbius transformation M = M (a,b,c,d) by M (x) := M (t(x)) = (ad bc)(x + ) (ax + b) + (cx + d), tr (M) := (a + d) /(ad bc), M := (a + b + c + d )/(ad bc). Proposition Let M be a Möbius transformation. Then M and min{m (x) : x R} = ( M M 4), max{m (x) : x R} = ( M + M 4). Proof: The equation M (x) = /ε for x has a solution iff its discriminant is nonnegative, i.e., if ε M ε +. The minimum and the maximum ε for which this holds are the solutions of ε M ε + =.

3 If tr (M) > 4, then M is hyperbolic, has a stable fixed point s R with M(s) = s, M (s) <, and an unstable fixed point u R with M(u) = u, M (u) >. If tr (M) = 4 then M is parabolic and has a unique fixed point s R with M (s) =. If tr (M) < 4 then M is elliptic and, unless it is the identity, has a unique fixed point in U and no fixed point in R. A rotation by α is the transformation R α = M (cos α,sin α, sin α,cos α ), R α (t) = t + α. A horocyle in U is a Euclidean circle tangent to the real axis or an Euclidean line parallel to the real axis. A horocycle in D is an Euclidean circle tangent to the unit circle. Definition The principal curve γ U of a Möbius transformation M which is not a rotation, is defined as follows. If M is hyperbolic, then γ is the geodesic which pases through u, i,s. If M is parabolic, then γ is the horocycle which passes through and s. If M is elliptic, then γ is the hyperbolic circle with hyperbolic center in s which passes through. All points M n (i) lie on the principal curve γ of M. The contracting interval of M and the expansing interval of M are defined by U(M) = {x R : M (x) < }, V(M) = {x R : (M ) (x) > }. For rotations we have U(R α ) = V(R α ) =. If M is not a rotation then both V(M) U(M) are nonempty intervals, M(U(M)) = V(M), U(M) > π, and V(M) < π. The values M(i) and M() of a Möbius transformation have a probabilistic interpretation. Given a compact metric space X, we denote by M(X) the space of Borel probability measures with weak topology, i.e., lim n ν n = ν iff lim n fdνn = fdν for each continuous function f. A continuous function F : X Y extends to a continuous function F : M(X) M(Y ) by (Fν)(U) = ν(f (U)). Denote by δ x the Dirac point measure concentrated on x, i.e., δ x (U) = iff x U. Measures on T and R which are absolutely continuous with respect to the Lebesgue measure have densities, which are nonnegative functions with unit integral. If F : X Y is a differentiable and increasing map and h is a density of a measure µ on X, then Fµ has density h(f (y)) (F ) (y). The uniform Lebesgue measure l on T has the constant probability density h(t) = /π. The Cauchy measure xl on R has density h(x) = /π( + x ). The densities h (a,b,c,d) of M (a,b,c,d) l on T and h (a,b,c,d) of xm (a,b,c,d) l on R are h (a,b,c,d) (t) = π h (a,b,c,d) (x) = M (a,b,c,d) (i) = µ + iσ = (ad bc)( + x (t)) (d x(t) b) + (c x(t) a) (ad bc)/π (dx b) + (cx a) = (ac + bd) + (ad bc)i c + d σ/π (x µ) + σ, where Thus h (a,b,c,d) is the density of the generalized Cauchy measure with parameters µ and σ. While Cauchy measures have infinite variance and no mean, the parameters µ and σ play a similar role as the mean and variance of the normal distribution. If X, X are random variables with generalized Cauchy distributions with parameters µ, σ and µ, σ respectively, then X + X has Cauchy distribution with parameters µ + µ, σ + σ. A measure µ M( D) can be characterized by its mean E(µ) := D zdµ 3

4 which is a complex number in the closed unit disc D. For a point measure we have E(δ t ) = e it which belongs to D. For generalized Cauchy measures we get E(M (a,b,c,d) l) = π π h (a,b,c,d)(t)e it dt which belongs to D. Proposition 3 If M is a MT, M the corresponding circle MT and M the corresponding disc MT, then E(Ml) = M() = d(m(i)). See Kůrka [8] for a proof. For M = M (a,b,c,d) we have M() = (d a) + (b + c)i (d + a) (b c)i, M() = M M + A Möbius transformation M is uniquely determined by its mean M() and its unit tangent vector M ()/ M () (see Katok [4]). We shall need several criteria for convergence of generalized Cauchy measures. Proposition 4 Let (M n : R R) n be a sequence of MT and x R. The following conditions are equivalent. () lim n M n (i) = x. () lim n M n () = d(x). (3) lim n M n l = δ t(x). (4) For each interval I x, lim n Mn (I) = π. (5) There exists c > and a sequence of intervals I m x such that lim I m =, m and m, liminf M n n (I m) > c Proof: () () follows from d(i) =. () (3) follows from Proposition 3. (3) (4) follows from the definition of convergence of measures. (4) (5) is trivial. (5) (): If Mn (I m) > c, there exists x R with (Mn ) (x) c/ I m. By Proposition, we get lim n M n =, and therefore lim n M n () =. There exist rotations R n such that lim n R n M n () = d(x), and we get lim n M n (R n (I m)) = π. Since liminf n M n (I m) c, we get lim n R n = Id and lim n M n () = lim n R n R nm n () = d(x). Proposition 5 Let (M n : R R) n be a sequence of Möbius transformations and assume that there exist distinct y, z R such that lim n M n (y) = lim n M n (z) = x. Then lim n M n (i) = x. If lim n a n /c n = lim n b n /d n = x, then lim n M (an,b n,c n,d n)(i) = x. Proof: Let M n = M (an,b n,c n,d n) and a n + b n + c n + d n =. Assume first x, y, z R. Then c n y + d n and c n z + d n are bounded away from zero, and lim n a n d n b n c n =. It follows lim M (a n d n b n c n )(y i) n(y) M n (i) = lim n n (c n y + d n )(c n i + d n ) = If some of the x, y, z are, the proof is analogous. 4

5 3. Möbius iterative systems For a finite alphabet A, denote by A := m Am the set of words of A and by A + := m> Am the set of nonempty words. With the concatenation operation, A + is the free semigroup over A. The length of a word u = u... u m A m is denoted by u := m. We denote by u [i,j) = u i... u j and u [i,j] = u i...u j subwords of u associated to intervals. We denote by A N the Cantor space of infinite sequences of letters of A equipped with metric d A (u, v) := k, where k = min{i : u i v i }. The shift map σ : A N A N is defined by σ(u) i = u i+. Given v A k, v A N is defined by (v ) km+j = v j. A subshift is a nonempty subset Σ A N, which is closed and σinvariant, i.e., σ(σ) Σ. For a subshift Σ there exists a set D A + of forbidden words such that Σ = S D := {x A N : u x, u D}. A subshift is uniquely determined by its language L(Σ) := {u A + : x Σ, u x}. Denote by L n (Σ) := L(Σ) A n. The cylinder of a word u L(Σ) is defined by [u] := {v Σ : v [, u ) = u}. An iterative system is an A + action over a compact metric space X, i.e., a continuous map F : A + X X satisfying F uv = F u F v, (the discrete topology is assumed on A + ). An iterative system is determined by its generators (F a : X X) a A. Definition 6 We say that F : A + R R, is a Möbius iterative system, if all (F a : R R) a A are orientationpreserving Möbius transformations. The convergence space X F A N and the naming map Φ : X F R are defined by X F := {u A N : lim j F u [,j) (i) R}, Φ(u) = lim j F u [,j) (i). Thus u X F iff the limit lim j F u[,j) (i) exists, and belongs to R. We denote by s u the stable fixed point of F u (provided F u is not elliptic) and by γ u the principal curve of F u (which passes through i and s u. We denote by U u := {x R : F u (x) < }, V u := {x R : (F u ) (x) > } the contracting interval of F u and the expansing interval of F u Proposition 7 Let F : A + R R be a Möbius iterative system. respectively. () For u A N, v A + we have vu X F iff u X F, and Φ(vu) = F v (Φ(u)). () For v A + we have v X F iff F v is not elliptic. Then Φ(v ) = s v is the stable fixed point of F v. Proof: () is trivial. (): All F v k(i) lie on the principal curve γ v. If F v is elliptic, then γ v is a hyperbolic circle, so F v k(i) cannot converge. If F v is hyperbolic or parabolic, then the stable fixed point s v attracts all points of U. Theorem 8 Let F : A + R R be a Möbius iterative system. () If {V u : u A + } is not a cover of R, then Φ(X F ) R. () If {V u : u A + } is a cover of R, then Φ(X F ) = R, there exists a subshift Σ X F on which Φ is continuous, and Φ(Σ) = R. 5

6 Proof: (): Assume that {V u : u A + } is not a cover of R and that x R does not belong to any V u. There exists an interval I x which is disjoint from all V u. Given u A N, then for each n we have Fu [,n) (I) I. By Proposition 4, F u[,n) l cannot converge to δ t(x) and therefore x Φ(X F ). (): Assume that {V u : u A + } is a cover of R. By compactness, there exists a finite set B A + such that {V u : u B} is a cover of R. We work now with the iterative system (F b : R R) b B. Let l > be a Lebesgue number of {V b : b B}, so that for each x R there exists b B such that the closed ball B l (x) = {y R : d(y, x) l} is a subset of V b. There exists an increasing continuous function ψ : [, ) [, ) such that ψ() =, ψ(t) < t for t > and for each b B and each interval I U b we have F b (I) ψ( I ). It follows that for each interval I V b we have F b (I) ψ ( I ). Given x R choose u B such that B l (x) V u, choose u B such that B l (Fu (x)) V u. We continue in this way and construct u B N. Denote by c := min{ V b : b B}. Then for each interval I x there exists n such that for all j > n we have Fu [,j) (I) min{ψ j ( I ), c, l}. By Proposition 4 we get u X F and Φ(u) = x, so we have proved Φ(X F ) = R. Let S B N be the set of all infinite words obtained in this manner. Then Φ : S R is surjective and S is shiftinvariant. We show that S is closed. For b B set W b = {x V b : B l (x) V b } and for u B n+ set W u := W u F u (W u ) F u[,) (W u ) F u[,n) (W un ). Then u S iff n W u [,n). This intersection is nonempty iff all W u[,n) are nonempty so S is closed. To prove that Φ is continuous on S, it suffices to show that for each u S, the intersection of all W u[,n) is a singleton. Given u S and m n we have Fu [,m] (W u[,n] ) Fu [,m] F u[,m) (W um ) = Fu m (W um ) U um, so W u[,n) = F u F u (W u[,n) ) ψ( F u (W u[,n) ) ) = ψ( F u F u [,] (W u[,n) ) ) ψ ( F u [,] (W u[,n) ) ) < ψ n+ ( F u [,n] (W u[,n) ) ) ψ n+ (π). Since ψ(t) < t and the only fixed point of ψ is zero, we get lim n W u[,n) =, so Φ is continuous on S. We return to the alphabet A and take the concatenation map c : B + A + which assigns to a word of B + the concatenation of its letters. We define Σ A N by Σ = {σ i (c(u)) : u S, i }. Then Σ satisfies all required properties. Definition 9 We say that (F, Σ) is a Möbius number system, if F : A + R R is a Möbius iterative system and Σ X F is a subshift such that Φ : Σ R is surjective and continuous. 4. Sofic subshifts A labelled graph over an alphabet A is a structure G = (V, E, s, t, h), where V is a finite set of vertices, E is a finite set of edges, s, t : E V are the source and target maps, and h : E A is a labelling function. A graph G is essential, if for each vertex q V there exist edges e, e E with t(e ) = q = s(e ). A finite or infinite 6

7 word u E + E N is a path in G if t(u i ) = s(u i+ ) for all i. The source and target of a finite path u E n are s(u) := s(u ), t(u) := t(u n ). We denote by O q E N the set of all infinite paths with source q. The label h(u) A + A N of a path u is defined by h(u) i := h(u i ). We denote by Σ G E N the subshift of all paths of G and by Σ G the subshift of all their labels. The languages of these subshifts are denoted by L G and L G. A subshift Σ is sofic iff there exists a labelled graph G such that Σ = Σ G. In this case we say that G is a presentation of Σ. Given q V, we denote by i(q) the longest common suffix of labels of all paths with target q, and by o(q) the longest common prefix of labels of all paths with source q. A subshift Σ is of finite type (SFT), if there exists a finite set D A + such that Σ = S D. We denote by L D := L(Σ D ) its language. Each SFT is sofic (see Lind and Marcus [9]). A finite set P L G is a suffix code for G, if each long enough finite path of G has a unique suffix which belongs to P. If i(q) λ for each q V, then {u L G : h(u) = i(t(u))} is a suffix code. A selector for G = (V, E, s, t, h) is a map K : V E which selects at each vertex an outgoing edge, so s(k(p)) = p. A selector K defines for each p V a unique path k p O q by k p = K(p), kp i+ = K(t(kp i )). If K is a selector, then k p is eventually periodic. If Φ : Σ G R is a factor map, the cylinder of p V is defined as [p] Φ = Φ(h(O p )). Theorem (Convergence theorem) Let F : A + R R be a Möbius iterative system and let G = (V, E, s, t, h) be a labelled graph. Assume that there exist closed intervals (I q ) q V and a finite suffix code P L G such that for each path u P we have I t(u) U h(u), and F h(u) (I t(u) ) I s(u). Then Σ G X F and Φ : Σ G R is continuous. Proof: There exists an increasing continuous function ψ : [, ) [, ) such that ψ() =, < ψ(t) < t for t >, and for every u P and any interval W U h(u) we have F h(u) (W) ψ( W ). There exists a constant d > such that for every u P, every suffix v of u and any interval W U h(u) we have F h(v) (W) d W. For a given infinite path u Σ G let Su n be the set of all words v (A + ) N such that v [,n) = u [,j) for some j, v k P for all k >, and v is a suffix of some v P (we do not distinguish a word in (A + ) N from its concatenation in A N ). Sets Su n+ Su n are nonempty and closed. By compactness there exists v n Sn u and for each n there exists j n such that v [,n) = u [,jn). Set W := I s(v), W n := F h(v[,n) )(I s(vn)). Since F h(vn)(i t(vn)) I s(vn) and t(v n ) = s(v n+ ), we get W n+ = F h(v[,n] )(I t(vn)) F h(v[,n) )(I s(vn)) = W n. For m < n we get by induction F h(v[m,n) )(I s(vn)) I s(vm) = I t(vm ) U h(vm ) and therefore W n d F h(v[,n) )(I s(vn)) d ψ( F h(v[,n) )(I s(vn)) ) < d ψ n ( I s(vn) )) d ψ n (π). Since the only fixed point of ψ is, we have lim j W n =, so there exists a unique x n W n. For m n we have (F h(v[,n) )) (W m ) = (F h(v[m,n) )) (I s(vm)) I s(vn), so liminf n (F h(v[,n) )) (W m ) c := min{ I q : q V } and therefore lim n F u[,n) (i) = x by Proposition 4. Thus h(u) X F and Φ(h(u)) = x, so Σ G X F. We show that Φ : Σ G R is continuous. Given w Σ G, let u Σ G be an infinite path with h(u) = w, let v n Sn u and let (j n) n be the sequence of 7

8 indices such that v [,n) = u [,jn). For m > let n be such that j n m < j n+. Then Φ(w) F h(u[,jn) )(U h(ujn )), and F h(u[,jn) )(U h(ujn )) dψ n (π) dψ m/p (π), where p := min{ u : u P }. Thus the diameter of Φ([w [,m) ]) is at most dψ m/p (π), so Φ : Σ G R is continuous. Theorem Let F : A + R R be a Möbius iterative system, let G = (V, E, s, t, h) be an essential labelled graph such that Σ G X F and Φ : Σ G R is continuous. Then there exist selectors L, R such that for their paths l p, r p O p we have [p] Φ J p := [Φ(h(l p )), Φ(h(r p ))]. Proof: For p V, the cylinder set [p] Φ = Φ(h(O(p)) is closed. Since G is assumed to be essential, there exist arbitrarily long paths u with target t(u) = p. We have F u ([p] Φ ) Φ[h(u)], and the diameter of this set goes to as the length of u goes to infinity. It follows that there exist unique a p, b p [p] Φ such that [p] Φ [a p, b p ] and the diameter of F u ([a p, b p ]) goes to zero as the length of u goes to infinity. Since for each p we have [p] Φ = {F h(e) ([q] Φ ) : p e q}, there exist an edge e = L(p) E such that p e q and a p = F h(e) (a q ). Analogously there exists an edge e = R(p) E with source p, target q and b p = F h(e )(b q ). Thus L, R are selectors and we have their paths l p, r p. For a given vertex p let m, n be the preperiod and period of l p and set q = t(l p [,m) ) = t(lp [,m+n) ), u := h(lp [,m) ), v = h(l p [m,m+n) ), so h(lp ) = uv. By Proposition 7, Φ(h(l p )) = F u (s v ), where s v is the stable fixed point of F v. For every k we have a p = F u (a q ) = F uv k(a q ) Φ([uv k ]), Φ(h(l p )) = F uv k(s v ) Φ([uv k ]). Since the diameter of Φ([uv k ]) converges to zero as k goes to infinity, we get Φ(h(l p )) = a p. Similarly we prove Φ(h(r p )) = b p. Proposition Let L, R be selectors such that for each selector K and for each p V we have Φ(h(k p )) [Φ(h(l p )), Φ(h(r p ))]. Assume moreover, that the length of intervals [Φ(h(ul p )), Φ(h(ur p ))] goes to zero as the length of a path u with target p goes to infinity. Then [p] Φ [Φ(h(l p )), Φ(h(r p ))]. The proof follows from Theorem. Since there is only a finite number of selectors for a given labelled graph, the left and right selectors L, R can be found effectively. We say that a continuous surjective map Φ : X Y has the extension property, if for any continuous map ϕ : X Y there exists a continuous map F : X X such that Φ F = ϕ. In this case, any continuous map G : Y Y can be lifted to a continuous map F : X X such that Φ F = G Φ. By a theorem of Weihrauch (see Weihrauch [], Theorem 3.., page 7 or Kůrka [6] Theorem 3.8, page ), for every compact metric space Y and any Cantor space X there exists a continuous surjection Φ : X Y with the extension property. X F X ϕ Y Φ 8 X F X Φ Y G Y Φ

9 Theorem 3 (Surjectivity theorem) Let F : A + R R be a Möbius iterative system, let G = (V, E, s, t, h) be an essential labelled graph such that Σ G X F and Φ : Σ G R is continuous. Let L, R be selectors from Theorem, and l p, r p their paths. Assume that the intervals J p := [Φ(h(l p )), Φ(h(r p ))] cover R and that for each p, the intervals {F h(e) (J q ) : p e q} cover J p. Then Φ(Σ G ) = R and [p] Φ = J p for each p V. If moreover the open intervals J p cover R and if {F h(e) (J q ) : p e q} is a cover of J p, then Φ : Σ G R has the extension property. Proof: By induction we get J p {F h(v) (J t(v) ) : v L n G, s(v) = p} for each n >. Since for v L n G the endpoints of interval F h(v)(j t(v) ) belong to Φ([h(v)]), we get diam(f h(v) (J t(v) )) diam(φ([h(v)])). Since Φ is assumed to be continuous, the diameter of these sets goes to zero as the length of v goes to infinity. Given x R, we construct an infinite word in Φ (x) as follows. Choose q V such that x J q v and choose an edge q v q such that x F h(v)(j q ). Then choose an edge q q such that F h(v (x) F ) h(v )(J q ), so x F h(vv )(J q ). Continuing in this manner, we get a path v Σ G such that x F h(v[,n) )(J qn ) for all n. Then the distance d(x, Φ(h(v [,n] ))) goes to zero as n, so x = Φ(h(v)). Thus Φ(Σ G ) = R and [p] Φ = J p for each p V. To show that Φ : Σ G R has the extension property, assume that W n is a sequence of closed intervals with W n+ W n and lim n W n =. There exists n and q V such that W q J q. There exists n > n and an edge v q q such that F h(v (V ) n ) F h(v)(j q ). We continue in this manner to construct v Σ G as the concatenation of all v i. 5. Arithmetical algorithms To perform arithmetical algorithms, we need Möbius transformations with coefficients in a computable field, and a finite automaton for the language of the subshift. A deterministic finite automaton over an alphabet A is a system A = (Q, δ, q ), where δ : Q A Q is a partial transition function and q Q is the initial state. Function δ is extended to a partial function δ : Q A Q by δ(q, λ) = q, and δ(q, ua) = δ(δ(q, u), a), where the lefthandside is defined iff the righthandside is defined. The language L(A) of A consists of words u A + which are accepted, i.e., for which δ(q, u) is defined. Given a sofic subshift Σ A N, the minimal automaton A Σ which accepts L(Σ) is constructed as follows. Define an equivalence on L(Σ) by u v iff w A +, (uw L(Σ) vw L(Σ)). The set Q of equivalence classes of is finite and A Σ := (Q, δ, {λ}), where δ(p, a) = q iff there exists u p such that ua q. We identify each q Q with some its element, so Q A and the initial state is q = λ. Assume now that (F, Σ) is a Möbius number system with sofic subshift Σ X F such that Φ : X F R is continuous and surjective, and let A Σ = (Q, δ, λ) be the minimal finite automaton for Σ. We define a system of intervals (J p ) p Q as follows. Consider a labelled graph G Σ = (Q, E, s, t, h) where E = {(q, a) Q A : δ(q, a) is defined}, s(q, a) = q, t(q, a) = δ(q, a), h(q, a) = a. The language of G Σ is L(Σ). The graph G Σ is not essential, in particular the initial vertex λ Q has no incoming edges. Let G = (Q, E, s, t, h) be the essential subgraph of G Σ, whose set 9

10 of vertices Q Q consists of all q Q such that there exist arbitrarily long paths u with target q. We use Theorem, to construct selectors L, R for G and intervals J p = [Φ(h(l p )), Φ(h(r p ))] for each p Q. For p Q \ Q we define intervals J p by induction. If p Q is such that J q are defined for all p e q, then let J p is a minimal interval which contains all F h(e) (J q ) for all edges p e q. Some of these intervals (for example that of λ) may be the full space R. We say that K is a computable field, if K is a countable subfield of R and the arithmetic operations of addition, subtraction, multiplication, division and the characteristic function of inequality are recursive. The field Q of rational numbers is computable. If K is a computable field, and if x,..., x n K are positive, then K( x,..., x n ) (the smallest subfield of R which contains K and all x i ) is a computable field. Given a computable field K, denote by I K the set of closed intervals [x, y] with x, y K = K { }. We allow intervals with x > y (which contain ) and intervals with x = y which stand for the full space R. On intervals we have addition given by [x, x ] + [y, y ] = {x +y : x [x, x ], y [y, y ]}, and similarly for other operations. If both I and J contain, then I + J = R. Assume that (F, Σ) is a Möbius number system with a sofic subshift Σ, such that the coefficients of F a belong to a computable field, and let A Σ = (Q, δ, λ) be the minimal automaton for Σ with the system of intervals (J q ) q Q. The endpoints a q, b q of J q can be obtained as solutions of quadratic equations, so they belong to a possibly larger computable field K. In K we can evaluate the operation of Möbius transformations on intervals F a ([x, y]) = [F a (x), F a (y)]. Then we have arithmetical algorithms analogous to those described in Vuillemin [] or Kornerup and Matula [5]. Given input words u, v A +, we compute their sum as follows. Set q u = δ(λ, u), q v = δ(λ, v), I u = F u (J qu ), I v = F v (J qv ), I w = I u + I v. We now find the output word w A as in the proof of Theorem 3. We initialize q w := λ, M w := Id, and we search e in a loop for q w q such that M w (I w) F h(e) (J q ). In this case we output h(e), update q w := q, M w := M w F h(e) and repeat the loop. The algorithm stops when no e, q with the required properties exist. The algorithm also works in an online manner (see Figure ) for infinite words u, v A N, whose sum it computes in an infinite loop and writes it to the output word w A N. We proceed analogously as in the case of finite input words, but we read a e new input letter only when the output loop cannot find any new state q w q. The new input letter is read for that argument, whose current interval I is longer. We can use the Euclidean length of intervals for this comparison, It may happen that the algorithm does not give any output, for example when the value of both u and v is. On the same principles we can construct algorithms for other arithmetic operations and also conversion algorithms between different Möbius number systems. 6. Semiregular continued fractions Regular continued fractions are based on iterations of transformations + x and /x. Given a positive number x, we repeatedly subtract one till we get a number in the interval (, ). Then we compute its reciprocal value /x and continue in the same way (see e.g. Perron []). Since the transformation /x is orientationreversing, we use rather F (x) = /x which represents the rotation of the unit circle by π. It is then natural to allow as coefficients also negative numbers, so we take transformations F (x) = + x, F (x) = + x. This leads to continued fractions of the form

11 function sum(u,v:input words; w:output word); var p u, p v, p w : states of A; a: letter of A; M u, M v, M w : matrices; I u, I v, I w : intervals; begin p u := λ; p v := λ; p w := λ; M u := Id; M v := Id; M w := Id; I u := M u (J pu ); I v := M v (J pv ); repeat I w := Mw (I u + I v ); If ( q, a)(q = δ(p w, a) & I w F a (J q )) then begin write a to w; p w := q; M w := M w F a ; end; else begin If I u > I v then begin read a from u; M u := M u F a ; p u := δ(p u, a); I u := M u (J pu ); end; else begin read a from v; M v := M v F a ; p v := δ(p v, a); I v := M v (J pv ); end; end; end; end; Figure. The online sum algorithm for Möbius number systems a a a with a i > for i >, which are known as semiregular continued fractions (see e.g., Perron [] or Dajani and Kraaikamp []). Given a number x, we subtract or add till we get a number in the interval (, ), so we have two possibilities. Then we apply /x and continue in the same way. In this process, words and never occur, so they should be avoided to ensure the convergence. We take a simpler subshift forbidding the words and. This corresponds to continued fractions with coefficients whose absolute value are at least. Definition 4 The Möbius system of semiregular continuous fraction (SRCF) consists of the alphabet A = {,, }, subshift Σ D with forbidden words D = {,,,, }, and transformations F (x) = + x, F (x) = /x, F (x) = + x, To prove convergence, we use Theorem applied to the labelled graph in Figure left, and suffix code {u L G : h(u) = i(t(u))}. There exist I q U i(q), such that for each p i(q) q, we have F i(q) (I q ) I p (see Table top). To show surjectivity, we use Theorem 3 applied to the labelled graph in Figure right. The selectors, their paths and intervals J p are given in Table bottom. We see that J p cover R, but their interiors do not. All edges p e q with source p have the same label h(e), and F h(e) (J p) is covered by these J q, so Φ : Σ D R is surjective. The means F u () in the unit disc

12 Figure. Graphs for semiregular continued fractions q i(q) I q U i(q) F i(q) (I q ) p [, ] [, ] [, ],, 4, 5 [, ] [, ] [, ],, 4, 5 [, ] [, ] [, ], 3 3 [, ] [, ] [, ],, 3, 4 4 [, ] [, ] [, ],, 3, 4 5 [, ] [, ] [, ], 3 p o(p) L, R h(l p ) h(r p ) h(e) J p F h(e) (J p) t(e), () () [, ] [, ], 5, () () [, ] [, ], 5 3, 3 () () [, ] [, ] 3 3 3, 4 () () [, ] [, ] 3, 4 4 5, () () [, ] [, ], 5 5, () () [, ] [, ] Table. Convergence and surjectivity of semiregular continued fractions can be seen in Figure 3. The points F ua () are joined to F u () by a horocycle if a = or a =. If a =, then F u () = F u (), so no new point is obtained, but the unit tangent vector is changed by π. At some F u (), words u are written in the direction of the unit tangent vector of F u. We can prove the convergence also by Proposition 5 using the continued fraction theory. Denote by Z := Z { }, and Z := {a Z N : i >, a i > } the space of infinite semiregular continued fractions. Define a coding map c : Z Σ D by { a c(a) := a a... if n, a n a... an if a n+ =, & j n, a j Here we adopt the convention a := a for a > and = λ. Given a Z, the convergents p n (a), q n (a) are defined by p (a) =, p (a) =, p (a) = a, p n (a) = a n p n (a) p n (a), q (a) =, q (a) =, q (a) =, q n (a) = a n q n (a) q n (a). Both p n (a) and q n (a) are increasing functions in each coordinate a i. We can prove by induction q n+ (a) > q n (a) > n, p n (a)q n+ (a) p n+ (a)q n (a) =, and p n (a) q n (a) p n (a) q n (a) = q n (a)q n (a) n.

13 / 3 3 3/ /3 /3 / / /3 /4 /4 /3 Figure 3. Semiregular continued fractions Since n n converges, p n (a)/q n (a) is a Cauchy sequence and has a limit, so for each a Z and each n > we get F c(a[,n] )(x) = p n (a) x + p n (a) q n (a) x + q n (a), p n (a) Φ(c(a)) = lim n q n (a). 7. Binary continued fractions The convergence in semiregular continued fractions is quite slow, so we add the transformation F (x) = x to make it faster. Several sofic subshifts can be constructed which yield a Möbius number systems with these transformations. Definition 5 The Möbius system of binary continuous fractions (BCF) consists of the alphabet A = {,,, }, the sofic subshift Σ whose three presentations are in Figure 4 top, and transformations F (x) = + x, F (x) = /x, F (x) = + x, F (x) = x We use the graph in Figure 4 top left to prove convergence. and the graph in Figure 4 top right to prove surjectivity (see Figure 4 bottom). The means F u () can 3

14 q i(q) I q F i(q) (I q ) [, ] [, ] [, ] [, ] [ +, ] [, ] 3 [, ] [, ] 4 [, ] [, ] 5 [, + ] [, ] p L, R h(e) J p F h(e) (J p), [, ] [, ], [, ] [, ] 3, 4 [, ] [, ] 3 3, 4 [, ] [, ] 4 3, 5 [, ] [, ] 5, [, ] [, ] Figure 4. Convergence and surjectivity of binary continued fractions be seen in Figure 3. The points F ua () are joined to F u () by a horocycle if a = or a = and by a geodesic if a =. We can also prove the convergence using the continued fraction theory. If u L Σ and u L G {,, } +, then u can be written as u = m a m a... m k a k, where m i > for i > and either all a i are positive or all a i are negative (we have n = n ). In particular, the last a k since is a forbidden word. Thus F u (x) = m (a + m (a + + m k (a k + x) ) = m (b + x) where m = m + + m k and b. If we write an infinite word u Σ as u u u..., then consecutive u i have different sign, so the absolute value of the limit of F u[,n) can be written as m m b + m m b + m b + = n b + n b + n b + where n = m and n i = m i n i. Since all m i are nonnegative, two consecutive n i, n i cannot be both negative. Thus we get an infinite continued fraction [c, c, c,...], where c j = nj b j are positive, and if c j <, then c j >. In standard continued fractions with positive coefficients c j, the convergents satisfy q j = c j q j + q j q j + q j, so they grow at least as fast as the Fibonacci sequence and therefore the continued fraction converges. In our case, if c j <, then q j+ q j + q j q j + q j. Thus if we leave out indices j for which c j <, the resulting convergents q j grow at least as fast as the Fibonacci sequence and the continued fraction converges. 4

15 / 3 3 3/ /3 / / /3 /3 /4 /4 /3 Figure 5. Binary continued fractions 8. Binary signed continued fractions The naming map of the binary continued fractions does not have the extension property, so we need a larger subshift to get it. In analogy to the binary signed system we allow words k and k for large enough k. Definition 6 The Möbius system of binary signed continuous fraction (BSCF) consists of the alphabet A = {,,, }, the subshift Σ D of finite type with forbidden words D = {,,,,,,,,,,,,, }, and transformations F (x) = + x, F (x) = /x, F (x) = + x, F (x) = x The convergence can be verified (with the help of a computer) with the graph whose vertices are L 8 (Σ). The surjectivity can be verifed using the presentation of Σ D in Figure 6 top left. In Figure 6 top right we see that {J p : p V } (thick lines) is an open cover of R and for each p V, {F h(e) (J q ) : p e q} (thin lines) is an open cover of J p. The factor map Φ : X F R has therefore the extension property, so the arithmetic algorithms work in the system. 5

16 3 8 9 A A A p o(p) L, R h(l p ) h(r p ) h(e) J p F h(e) (J p) t(e), 3 () () [, 3 ] [, 3], 3 8, 4 () () [, 3] [, ] 4, 8, 3 () () [, 6] [, 3], 3 3, A () () [, 3] [, 4], A 4 7, 5 () () [ 3, ] [3, ] 5, 7 5, 8 () () [ 3, ] [, ], 8 6 7, 5 () () [6, ] [3, ] 5, 7 7 A, 6 () 3() [3, ] [4, ] 6, A 8 9, 9 () () [, ] [, ] 9 9, 6 () () [, ] [, ], 6, 9 A 9, 9 () () [4, 4] [, ] 9 Figure 6. Surjectivity of binary signed continued fractions Acknowledgments The research was supported by the Research Program CTS MSM I thank Alexandr Kazda for valuable comments. References [] M. F. Barnsley. Fractals everywhere. Morgan Kaufmann Pub., 993. [] K. Dajani and C. Kraaikamp. A note on the approximation by continued fractions under extra condition. New York Journal of Mathematics, 3A:69 88, 998. [3] G. A. Edgar. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. SpringerVerlag, Berlin, 99. [4] S. Katok. Fuchsian Groups. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 99. [5] P. Kornerup and D. W. Matula. An algorithm for redundant binary bitpipelined rational arithmetic. IEEE Transactions on Computers, 39(8):6 5, August 99. [6] P. Kůrka. Topological and symbolic dynamics, volume of Cours spécialisés. Société Mathématique de France, Paris, 3. [7] P. Kůrka. Iterative systems of real Möbius transformations. 8. [8] P. Kůrka. A symbolic representation of the real Möbius group. Nonlinearity, :63 63, 8. [9] D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 995. [] O. Perron. Die Lehre von Kettenbrüchen. Teubner, Leipzig, 93. 6

17 / 3/ /3 /3 / / /3 /4 /4 /3 Figure 7. Binary signed continued fractions [] J. E. Vuillemin. Exact real computer arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):87 5, August 99. [] K. Weihrauch. Computable analysis. An introduction. EATCS Monographs on Theoretical Computer Science. SpringerVerlag, Berlin,. 7