The van der Corput embedding of ax + b and its interval exchange map approximation

The van der Corput embedding of ax + b and its interval exchange map approximation Yuihiro HASHIMOTO Department of Mathematics Education Aichi University of Education Kariya 448-854 Japan Introduction In 4] we introduce interval preserving map approximation of a linear map 3x+ to attac the well-nown and still unsolved 3x + problem which is firstly proposed by Lothar Collatz in 930 s: Conecture Consider a map f : N N such that { 3n + f(n) = n/ if n is odd if n is even Then for each natural number n there exists a finite number t such that f t (n) = f f f(n) = }{{} t times In the investigations3]4] we use the binary van der Corput embedding β of natural numbers into unit interval 0 ] which gives a low-discrepancy sequence over 0 ] firstly introduced by van der Corput in 935(cf7] pp 9) and construct a right continuous biection F over 0 ) conugate to 3x + by way of β In this note we investigate a right continuous biection F over 0 ) conugate to the linear function ax + b its finite bit approximations which bring dynamics of interval exchange maps on 0 ) and observe substitution dynamics and transducers associated with the approximations As a result we obtain a necessary and sufficient condition for the minimality of the dynamics (0 ) F ) (Theorem 38) The binary van der Corput embedding of natural numbers and a conugacy of ax + b We follow the notations in 4] Definition Let n = g + g + + g 0 (= (g g g 0 ) for short ) be a binary expansion of a natural number n Then we define β : N 0 ) given by β(n) = g 0 + g + + g + = (0g 0g g ) As β : N 0 ) is one-to-one and β embeds natural numbers densely into 0 ) we call β the binary van der Corput embedding For a binary expression n = (g l g l g 0 ) we put l = ord(n) := log n] where x] stands for an integer not greater than x and n denotes an upper cut off of n at -th order; n = (g g g 0 ) n (mod ) To mae free from ambiguity we always tae the finite binary expression β(n) = (0g 0 g g l ) of any natural number n For a real number x 0 ) and N x denotes a cut off of x at ( )-th order in the binary expression; x = (0e e e ) = = e if x = (0e e ) = We use the notation x ) as a segment x x + ) hence a natural decomposition of segments () (0e e e ) ) = (0e e e 0) ) + (0e e e ) ) + = e

The van der Corput embedding of ax + b where a b) b c) stands for a division of an interval a c) at b Then the -th refinement of the unit interval 0 ) is given as () 0 ) = where {ν () =0 ) β(ν () ) = 0 } = {0 } with β(ν () i ) < β(ν () ) if and only if i < By definition we easily to see the followings Lemma For x y 0 ) and l m n N () x x ) () x ) y ) holds if and only if x = y and hence x ) = y ) (3) if x y there exists l N such that x ) l y ) l = (4) x ) + x ) and x ) = {x} = (5) β(m) = β(m ) thus β(m) ) = β(n) ) if and only if m = n (6) If l > ord(m) then β(m + n l ) = β(m) + β(n) l (7) (n ) l = n min{l} Let us consider a linear function f(n) = an + b on natural numbers where a b N and a > b We are to construct the conugacy F : 0 ) 0 ) of f : N N that is F β(n) = β f(n) holds for any natural number n Lemma 3 Suppose a is odd number Then we have () β f(m) ) = β f(n) ) holds if and only if β(m) ) = β(n) ) for any m n N () {β f(n) n N} is dense in 0 ) Proof () By Lemma (5) β(m) ) = β(n) ) means m n (mod ) which equivalents to am + b an + b (mod ) as a and are coprime hence β(am + b) ) = β(an + b) ) () Given x 0 ) and N consider m = β (x ) and tae b N with b b m (mod a) As a and is coprime we have φ(a) = ( ) φ(a) (mod a) hence m + b φ(a) m + b b (mod a) where φ stands for Euler s totient function Thus there exists n N with m + b φ(a) = an + b and hence β(an + b) = β(m + b φ(a) ) = x + β(b ) φ(a) x ) As a result for any x 0 ) and N there exists n N with x β(an + b) < Now we suppose a is odd Then the conugacy F is defined on the dense set β(n) by F (β(n)) := β(f(n)) and its extension to 0 ) is given as follows For any x 0 ) consider a sequence n = β (x ) = Obviously we see n + n (mod ) hence an + + b an + b (mod ) It follows from Lemma (4) that holds and then = β(an + + b) ) + β(an + + b) ) = β(an + b) ) β(an + b) ) consists of a unique point denoted by lim β(an + b) Thus we define the conugacy F : 0 ) 0 ) of the linear function f: (3) F (x) = lim β ( aβ (x ) + b ) Lemma 4 For x y 0 ) we have

Yuihiro HASHIMOTO 3 () F (x ) ) = F (x) ) hence F (x) = F (x ) () F ( x ) ) F (x) ) hence F is right continuous (3) F (x) ) = F (y) ) if and only if x ) = y ) hence F is inective Proof () By definition we have {F (x)} = β(aβ (x ) + b) ) β(aβ (x ) + b) ) = F (x ) ) = that is F (x) F (x ) ) while F (x) F (x) ) by Lemma () Thus we see F (x ) ) F (x) ) which means F (x ) ) = F (x) ) by Lemma () () y x ) means y = x It follows from () that F (y) F (y) ) = F (y ) ) = F (x ) ) = F (x) ) For any ε > 0 tae N with < ε Then for any y with x y x + (+) we see y x ) and F (y) F ( x ) ) F (x) ) that is F (y) F (x) < < ε showing the right continuity of F (3) Put m = β (x ) and n = β (y ) then we see F (x) ) = F (x ) ) = β f(m ) ) and F (y) ) = F (y ) ) = β f(n ) ) by () It follows from Lemma 3 () that F (x) ) = F (y) ) if and only if x ) = β(m ) ) = β(n ) ) = y ) Suppose that x y then x ) y ) = holds for some Then we have F (x) ) F (y) ) = hence F (x) F (y) Lemma 4 () is an extension of a natural property (4) f(n ) = f(n) of the linear function f(n) = an + b with a b N while ()(3) implies that F brings segment-wise exchange over the refinement () of the unit interval in any order Indeed we have the following Proposition 5 For any N and x 0 ) the conugacy F gives a right continuous biection F : x ) F (x) ) Particularly F preserves the Lebesgue measure µ on 0 ) Proof All we have to show is surectivity of F For any y 0 ) and l N there exists n l N with F β(n l ) = β f(n l ) y ) l by Lemma 3 () By Lemma 4 we see F ( β(n l ) ) l ) F (β(n l )) ) l = y ) l Let x be the unique accumulation point {x} = β(n l ) ) l Then F (x) F (β(n l )) ) l y ) l holds for any l l meaning that F (x) y ) l = {y} l hence y = F (x) and F is a surection 0 ) 0 ) Combining Lemma 4 ()(3) and surectivity of F we see F ( x ) ) = F (x) ) for any N Note that µ( x ) ) = µ( F (x) ) ) = It is shown that the σ-field generated by the segments x ) N x 0 ) coincides with the Lebesgue measurable sets Hence the assertion

4 The van der Corput embedding of ax + b Note that for any natural number n β(n) ) = β(n) β(n) + ) if > ord(n) thus w β(n) ) means w β(n) Therefore we state the right continuity of F only Indeed for f(n) = n + we see lim w (0) w<(0) F (w) = (0) (000) = F ((0) ) 3 Segment-wise linear approximation of the conugacy F In view of Proposition 5 we construct a sequence of segment-wise linear functions F s approximating the conugacy F Definition 3 For each N we define the -th approximant F as If x β(n) ) we see F (x) = x x + F (x) F (x) = x β(n) + F (β(n)) thus F ( β(n) ) ) = F β(n) ) compatible with Proposition 5 F β(n) ) for any x β(n) ) means F (x) F (x) < Moreover the fact F (x) F (x) for any x β(n) ) and then for any x 0 ) Therefore the sequence F = approximates F uniformly on 0 ) so F simulates the behavior of F that exchange the segments β(n) ) s We also note that (3) is expressed as F (x) = lim F (x) In the following subsections we investigate the approximants F s to extract dynamical characteristics of the map f(n) = an + b 3 Behavior of carries and exchange of segments For each N we define an integer valued function then we see f(n ] ) an ] + b τ (n) = = (3) f(n ) = τ (n) + f(n) Namely the function τ describes the amount of the carry from the lower bits in the calculation n an+b Theorem 3 Suppose that a is odd and 0 b < a Then for n N we have () τ (n) {0 a } () For binary expressions n = (g p g 0 ) and f(n) = (h q h 0 ) we have and where g = 0 for > p h = ( g + τ (n) ) = q ] ag + τ (n) τ + (n) = (3) Let (0g 0 g g p ) be the binary expression of β(n) Then the image of the segment β(n) ) = (0g 0 g ) ) = (0g 0 g 0) ) + (0g 0 g ) ) + by F is given as ) F ((0g 0 g 0) ) ) + F ((0g 0 g ) ) ) + if τ (n) is even F ( β(n) ) = F ((0g 0 g ) ) ) + F ((0g 0 g 0) ) ) + if τ (n) is odd

Yuihiro HASHIMOTO 5 Proof () As n and a > b we have an ] + b a ] + b a 0 = a a b ] a () By definition we see n + = (g g g 0 ) = g + n We also see f(n) + = h + f(n) while by (3) we have f(n) + = f(n + ) ( ) + = a(g + n + ( ) ) + b = ag + f(n + (3) ) ( = (ag + τ (n)) + f(n) ) + ( = ag + τ (n) ) + f(n) As a is odd we obtain the equation h = ( ag + τ (n) ) = ( g + τ (n) ) Again by (3) and f(n) + = h + f(n) we see while as n + = g + n we have f(n + ) = τ + (n) + + f(n) + = (τ + (n) + h ) + f(n) f(g + n ) = ag + f(n ) = (ag + τ (n)) + f(n) hence τ + (n) + h = ag + τ (n) Since 0 h / < and τ + (n) is an integer we come to τ + (n) = τ + (n) + h ] ] ag + τ (n) = (3) It comes from (3) that β (f(n)) + = β (f(n)) + + β(g + τ (n)) Then we have (g + τ (n)) = g whenever τ (n) is even and (g + τ (n)) = g whenever τ (n) is odd Thus we see ( ) ( ) F β(n) + F β(n + ) + = + if τ (n) is even ( ) F β(n) + if τ (n) is odd hence the assertion 3 Substitution dynamics induced by the conugacy F Definition 33 For each approximation order we label the segment β(n) ) as L (n) := τ 0 (n)τ (n) τ (n) We say the segment β(n) ) is painted in color τ (n) By Theorem 3 () the label L (n) is a string over the alphabet A = {0 a } with length + Note that τ 0 (n) = b by definition thus the original unit interval 0 ) is labeled as b As the approximation order increases each segment is divided into two segments: ) (33) β(n) ) = β(n ) ) β(n ) )+ β(n + ) Accordingly the refinements β(n ) ) + and β(n + ) ) are labeled respectively as + ] L + (n τ (n) ) = τ 0 (n) τ (n) (34) ] a + L + (n + τ (n) ) = τ 0 (n) τ (n) by Theorem 3 () +

6 The van der Corput embedding of ax + b Consider a free monoid A = =0 A over A with the concatenation as a multiplication operation and the empty string ε as a unit where A = {x x x x i A} and A 0 = {ε} In view of (33) and (34) we define a substitution ζ : A A as ζ : x x ] a + x which simulates the way to repaint the segments that the refinement (33) causes: ] ] τ (n) τ + (n ) τ + (n + τ (n) a + τ (n) ) = Consequently the refinement of the unit interval 0 ) (00) ) (0) ) (000) ) (00) ) (00) ) (0) ) induces an orbit of ζ with initial state b: ] ] ] b a + b b/] b ζ(b) = ζ (b) = Summing up we have the following a + b/] ] ] a + (a + b)/] ] ] a + (a + b)/] Proposition 34 The -th refinement () of the unit interval is painted in the pattern ζ (b) 33 A finite state transducer which represents ax + b Theorem 3 () is also expressed in terms of a finite state automaton Let T ab = (A I O {b} {0} δ) be a deterministic finite state transducer with states A an input alphabet I = {0 } an output alphabet O = {0 } an initial state b a final state 0 and a transition function δ : A I A O According to Theorem 3 () we define the transition rule as ( ] ) ag + x g if x is even (35) δ(x g) = ( ] ) ag + x g if x is odd Corollary 35 Given the binary expressions (g p g 0 ) of a natural number n define a sequence in A O (x i+ c i+ ) = δ(x i g i ) i = 0 q where x 0 = b q = p + ord(a) + and g i = 0 if i > p Then we have and the label of the segment β(n) ) is given as f(n) = (c q c q c 0 ) for each = 0 L (n) = x 0 x x Proof Let (h q h 0 ) be the binary expression of f(n) By definition we see x i+ = (ax i + g i )/] Recall that τ 0 (n) = b for any n N It comes from Theorem 3 () that { τ i (n) = x i and h i = (g i + x i ) g i if x i is even = g i if x i is odd hold hence c i = h i

Yuihiro HASHIMOTO 7 34 Minimality of the dynamical system (0 ) F )) As we have seen for each approximation order the approximant F exchanges the segments β(ν () ) s in the refinement () and hence induces a permutation π over {0 } satisfying F ( β(ν () ) ) ) = β(ν () π () ) ) or equivalently ( ν () π () = aν () + b) Note that as the approximation order increases we see (36) ν (+) = ν () and ν (+) + = ν() + by definition of the refinement () Combining Theorem 3 (3) and (36) we have whenever τ (ν () ) is even and ν (+) π + () = ν() π () = ν(+) π () ν (+) π + () = ν() π () + = ν (+) π ()+ and ν (+) π + (+) = ν() π () + = ν (+) π ()+ and ν(+) π + (+) = ν() π () = ν(+) π () whenever τ (ν () ) is odd As a result we obtain relations between the permutation π and π + ) Lemma 36 If the color τ (ν () ) of the segment β(ν () ) is even we have and if τ (ν () ) is odd π + () = π () and π + ( + ) = π () + π + () = π () + and π + ( + ) = π () Let π = π π π m be a cycle decomposition determined uniquely up to the order of π i s For each cyclic permutation π i = (p p p l ) consider the maximum subset Q {p p l } such that τ (ν () is odd if and only if q Q For a cycle π i = (p p p l ) let π +i be the permutation defined as Lemma 36 that is ( π +i (p + η) = π i (p ) + τ (ν p () ( ) + η) = p+ + τ (ν p () ) + η) η {0 } where l + is understood as q ) Proposition 37 π +i is a cyclic permutation whenever #Q is odd and π +i permutations whenever #Q is even consists of two cyclic Proof Put P 0 = {p p p l } and P = {p + p + p l + } Consider an orbit x = p x = π +i (x ) x 3 = π +i (x ) and a map ϕ(x ) = η if x P η Notice that x sl+ {p p + } for = l and s N by definition Then we see ϕ(x sl++ ) = ϕ(x sl+ ) whenever p / Q and ϕ(x sl++ ) = ϕ(x sl+ ) whenever p Q hence for each = l and s N x sl++ = π +i (x sl+) = p + + ϕ(x sl++ ) Suppose that #Q is even It is seen that ϕ(x l+ ) = ϕ(x ) = 0 hence x l+ = p + ϕ(x l+ ) = p = x namely the cycle (x x x l ) is a factor of π +i Considering another orbit y = p + y = π +i (y ) a similar argument shows the cycle (y y y l ) is also a factor of π +i Consequently we obtain a decomposition π +i = (x x x l )(y y y l )

8 The van der Corput embedding of ax + b Suppose that #Q is odd then we see ϕ(x l+ ) = ϕ(x ) = hence x l+ = p + ϕ(x l+ ) = p + = y showing that π +i is a cycle of the length l: π +i = (x x x l y y y l ) We are to find a condition that the permutation π induced by the approximant F becomes a cycle in any approximation order For a string w = w w w l A we put S(w) = l i= w i Note that the parity of S(w) equals that of the number of entries w i s which are odd In view of Proposition 37 the original unit interval 0 ) must be painted an odd color at least namely b = τ 0 (0 )) is odd Then S(ζ(b)) also should be odd We have assumed that a is odd Putting a = m+ we have S(ζ(b)) = ] b + ] a + b = m + + as (b + )/] = b/] + hence m should be even; a (mod 4) Conversely we have the following Theorem 38 The permutation π associated with the approximant F is a cyclic permutation in any approximation order hence the dynamical system (0 ) F ) is minimal if and only if a (mod 4) and b is odd Proof The necessity for b being odd and a (mod 4) is shown above Suppose a = 4m + and b being odd It comes from Theorem 3 () that S(ζ(x)) = m + x/] if x is even and S(ζ(x)) = m + + x/] if x is odd Then S(ζ(w)) is odd if and only if S(w) is odd for any w A As a result we see that S(ζ (b)) is odd for any hence π is a cyclic permutation by Proposition 37 which means for any segment x ) the orbit x ) F ( x ) ) F ( x ) ) F ( x ) ) covers 0 ): (37) i=0 F i ( x ) ) = 0 ) Tae any x 0 ) (37) shows that for any y 0 ) there exists 0 t with y F t ( x ) ) while F t (x) F t (x) ) = F t ( x ) ) Thus we have y F t (x) < Consequently the orbit {F s (x) s N} ] b is dense in 0 ) for any x 0 ) namely the dynamical system (0 ) F ) is minimal This result is contrast to Keane condition6]6]: no left endpoint of any segment is mapped to another ones which brings the minimality of the dynamics of interval exchange maps In our case the left endpoint x of any segment x ) is always mapped to another ones by F Example 34 (The case 3x+) This is the original case of Collatz The substitution ζ and the transducer T 3 are illustrated in Figure and respectively As 3 (mod 4) the dynamics (0 ) F ) is not minimal Indeed Table shows that permutations π s associated with F s are decomposed in two cycles A typical orbit of F 5 the approximation order = 5 is illustrated in Figure 5 π 0 (0) (0)() (0 )( 3) 3 (0 4 5)( 7 3 6) 4 (0 8 9 3 0)(4 4 6 5 5 7 3) Table Permutations π associated with 3x +

Yuihiro HASHIMOTO 9 0 0 b= 0 0 0 0 /4 / 3/4 Figure The color pattern of refinement induced by substitution ζ 0/0 / / /0 0 0/ 0/0 Figure Transitive diagram of T 3 Example 34 (The case 5x + ) The substitution ζ and the transducer T 5 are illustrated in Figure 3 and 4 respectively As 5 (mod 4) the conugacy F of f(x) = 5x + induces a minimal dynamical system on 0 ) Actually Table shows that permutations π s associated with F s are always cyclic A typical orbit of F 5 the approximation order = 5 is illustrated in Figure 6 0 0 b= 0 3 0 3 0 /4 / 3/4 Figure 3 The color pattern of refinement induced by substitution ζ 3 4 4 π 0 (0) (0 ) (0 3) 3 (0 4 3 7 5 6) 4 (0 8 6 5 3 4 3 9 7 4 0 5 ) Table Permutations π associated with 5x + 0/0 / 0/0 0 4 0/ 0/0 0/ / /0 3 /0 Figure 4 Transitive diagram of T 5 / Figure 5 Approximant F 5 3x + and a typical orbit for Figure 6 Approximant F 5 5x + and a typical orbit for

0 The van der Corput embedding of ax + b 4 Remars Notice that F preserves the Lebesgue measure µ on 0 ) by Proposition 5 therefore it will be a worthwhile question whether the minimality implies the ergodicity in our case However I have no statement about the ergodicity of F at this moment In 3] and 4] we have discussed the van der Corput embedding of the Collatz procedure { x x 0 /) G(x) = F (x) = lim β(3β (x ) + ) x / ) Then to solve the original Collatz conecture we are to prove the statement: for each n N there exists a finite number t with G t (β(n)) = / Note that G does not preserve intervals and so the Lebesgue measure The finite bit approximation G (x) = { x + x for x 0 /) F (x) for x / ) leads us to a mild version of the Collatz conecture a finite combinatorial problem proposed in 4] of which research is in progress Problem 4 (3x + problem on G ) Show that for any x 0 ) there exists t N such that ) ) G t (x) 0 ) β() ) = 0 / / / + / References ] K Daani and C Kraaiamp Ergodic theory of numbers Carus Mathematical Monographs 9 Mathematical Association of America 00 ] K Falconer Fractal Geometry: Mathematical Foundations and Applications nd ed Wiley 003 3] Y Hashimoto A fractal set associated with the Collatz problem Bull of Aichi Univ of Education Natural Science 56 pp -6 007 4] Y Hashimoto Interval preserving map approximation of 3x + problem Bull of Aichi Univ of Education Natural Science 6 pp 5-4 0 5] J Hopcroft R Motwani and J Ullman Introduction to Automata Theory Languages and Computations nd Ed Addison- Wesley 00 6] M Keane Interval Exchange Transformations Math Z 4 pp 5-3 975 7] L Kuipers H Niederreiter Uniform distribution of sequences Dover Publications 005 8] J Lagarias The Ultimate Challenge: The 3x + Problem AMS 0 9] M Lothaire Algebraic combinatorics on words Encyclopedia of mathematics and its applications 90 Cambridge University Press 00 0] F Oliveira and F L C da Rocha Minimal non-ergodic C -diffeomorphisms of the circle Ergod Th & Dynam Sys no 6 pp 843-854 00 ] M Queffélec Substitution Dynamical Systems - Spectral Analysis Lect Notes in Math 94 nd Ed Springer 00 ] A Rényi Representations for real numbers and their ergodic properties Acta Math Acad Sci Hungar 8 pp 477-493 957 3] M Shirvani and T D Rogers On Ergodic One-Dimensional Cellular Automata Commun Math Phys 36 pp 599-605 99 4] G Wirsching The Dynamical System Generated by the 3x + Function Lect Notes in Math 68 Springer 998 5] H Xie Grammatical complexity and one-dimensional dynamical systems Directions in chaos 6 World Scientific 997 6] J C Yoccoz Continued Fraction Algorithms for Interval Exchange Maps: an Introduction in Frontiers in Number Theory Physics and Geometry I pp 403-438 005