STRUCTURE THEOREM FOR (d,g,h)-MAPS 6 0 A.V.KONTOROVICHANDYA.G.SINAI 0 2 Abstract. The (3x+1)-Map, T,acts on the set, Π, of positiveintegers not n divisibleby2or3. ItisdefinedbyT(x)= 3x+1,wherekisthelargestinteger a 2k forwhichT(x)isaninteger. The(3x+1)-Conjectureasksifforeveryx∈Π J there exists an integer, n, such that Tn(x) = 1. The Statistical (3x+1)- 5 Conjectureasksthesamequestion,exceptforasubsetofΠofdensity1. The 2 Structure Theorem proven in [S] shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x+1)-Conjecture may be ] T true. Inthispaper,wepresenttheanalogoustheoremforsomegeneralizations of the (3x+1)-Map, and expand on the consequences derived in [S]. The N generalizationsweconsideraredeterminedbypositivecoprimeintegers,dand . g,withg>d≥2,andaperiodicfunction,h(x). ThemapT isdefinedbythe h formulaT(x)= gx+h(gx),wherekisagainthelargestintegerforwhichT(x) t dk a is an integer. We prove an analogous Structure Theorem for (d,g,h)-Maps, m and that the probability distribution corresponding to the density converges [ to the Wiener measure with the drift logg− d−d1logd and positive diffusion constant. Thisshowsthatitisnaturaltoexpectthattypicaltrajectoresreturn 1 totheoriginiflogg− d logd<0andescapetoinfinityotherwise. d−1 v 2 2 6 1 1. Introduction 0 1.1. The (3x+1)-Map and (3x+1)-Conjecture. Recall the definition of the 6 0 (3x+1)-Map, (see [L]). Take an integer x > 0, with x odd. Then 3x+1 divides / 2, so we can find a unique k >0 such that y = 3x+1 is again odd. In this way, we h 2k get a mapping T : x y defined on the set Π of strictly positive numbers not t a divisible by 2 or 3. W7−r→ite Π = 6Z++E, where E = 1,5 , is the set of possible m { } congruence classes modulo 6. : Foreveryinteger,x,with0d 2,andaperiodicfunction,h(x),satisfying: ≥ (1) h(x+d)=h(x), (2) x+h(x) 0(modd), ≡ (3) 0< h(x) 1. For m > 1, the induction goes as follows. To know which x have a given m-path, (k ,...,k ), we first assume we know the answer for 1 m the (m 1)-path, (k ,...,k ). 1 m−1 − Let k =k +k +...+k , and assumeby the induction hypothesis that there 1 2 m−1 are(d 1)m−1valuesforthetriplet(q ,r ,δ )whichsatisfyourequations. m−1 m−1 m−1 − Fix one such triplet, pick any integer, p , and set x=dg dkp +q +ε, m−1 m−1 m−1 and y = dg gm−1p +r +δ . Then we have γ (cid:0)(x) = (k ,...,k(cid:1) ), m−1 m−1 m−1 m 1 m−1 and y =Tm−(cid:0)1(x). Here we writ(cid:1)e pm−1 instead of just p to distinguish from the p we will have in the next paragraph. The triplet (q ,r ,δ ) is still gotten m−1 m−1 m−1 independently of p . m−1 We can alternatively break x into x = dg dk+kmpm+qm +ε for some qm < dk+km and also write z = Tm(x) = T (y) = d(cid:0)g t+δm, with(cid:1)t = gms+rm. The · key idea is to find the d 1 possible values for δ E, and with each we solve for m − ∈ the corresponding q and r , knowing q , r , and δ . We will again see m m m−1 m−1 m−1 that p =s and that (q,r,δ) do not depend on this value. m Since z = T (y), by assumption, we have dkmz = gy +h(gy), (again let h = h(gy)=h(gδ )) which expands to: m−1 dkm+1gm+1s+dkm+1grm+dkmδm =dgm+1pm−1+g2drm−1+gδm−1+h. (2.4) Remembering the two expressions for x, and setting pm = dkmp1+p2 (with 0 ≤ p2

Structure Theorem for (d,g,h)-Maps

Similar Structure Theorem for (d,g,h)-Maps