Regularity of some class of nonlinear transformations
aa r X i v : . [ m a t h . D S ] A ug Regularity of some class of nonlineartransformations
October 29, 2018
N. N. Ganikhodjaev Department of Mechanics and Mathematics,National University of Uzbek-istan, Vuzgorodok, 700095, Tashkent, Uzbekistan and Centre for Compu-tational and Theoretical Sciences, Faculty of Science, International IslamicUniversity Malaysia, 53100 Kuala Lumpur, Malaysia .M. R. B. Wahiddin Centre for Computational and Theoretical Sciences, Faculty of Science, In-ternational Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia [email protected] [email protected] Department of Mechanics and Mathematics,National University of Uzbek-istan, Vuzgorodok, 700095, Tashkent, Uzbekistan
Abstract
In this paper we consider quadratic stochastic operators designedon finite Abelian groups. It is proved that such operators have theproperty of regularity.
Mathematics Subject Classification : 37A25 , 37N25 , 46T99 ,47H60.
Key words :Quadratic stochastic operators; Finite Abeliangroup ; Regularity ; Ergodicity .
Let S n − = { x = ( x , · · · , x n ) ∈ R n : n X i =1 x i = 1 , x i ≥ ∀ i = 1 , · · · , n } (1)be the ( n − R n . The transformation V : S n − → S n − is called a quadratic stochastic operator (q.s.o.), if( V x ) k = n X i,j =1 p ij,k x i x j (2) [email protected] here p ij,k ≥ ,p ij,k = p ji,k , n X k =1 p ij,k = 1 (3)for arbitrary i, j, k ∈ { , · · · , n } . Such operators have applicationsin mathematical biology, namely theory of heredity, where the coeffi-cients p ij,k are interpreted as coefficients of heredity [1-3].Assume { V k x : k = 0 , , · · · } is the trajectory of the initial point x ∈ S n − , where V k +1 x = V ( V k x ) for any k = 0 , , · · · Definition 1
A q.s.o. V : S n − → S n − is called ergodic (respectivelyregular) if for any initial point x ∈ S n − the limit lim k →∞ k k − X i =0 V i x (respectively the limit lim k →∞ V k x ) exists. Evidently, any regular q.s.o. V has the ergodic property, but theconverse is not necessarily true.To determine whether some q.s.o. is ergodic or regular is rathercomplicated problem.S. Ulam in [4] presupposed the assumption that any q.s.o. V isergodic. Later M. Zakharevitch [5] showed, that this is an incorrect ypothesis in general. More precisely, he proved that the q.s.o. V, which is defined on the simplex S = { ( x, y, z ) : x, y, z ≥ , x + y + z = 1 } by the formula V : ˆ x = x + 2 xy ;ˆ y = y + 2 yz ;ˆ z = z + 2 xz. is not an ergodic q.s.o. and respectively is not regular.Later in [6] necessary and sufficient conditions were established forthe ergodicity of the so-called Volterrian q.s.o.: V : ˆ x = x (1 + ay − bz );ˆ y = y (1 − ax + cz );ˆ z = z (1 + bx − cy )where a, b, c ∈ [ − , . For a, b, c = 1 we have the initial example ofZakharevitch.
Theorem 1
Any q.s.o. in the above form is non-ergodic if and onlyif the three parameters a, b, c have the same sign. elow we’ll construct one class of q.s.o. and prove that all suchq.s.o. are regular and respectively ergodic. Let G be a finite Abelian group and S ( G ) be a set of all probabilisticmeasures on G. It is evident, that if | G | = n, then S ( G ) coincides with S n − (1) . Let further H ⊂ G be a subgroup of G and { g + H : g ∈ G } bethe cosets of H in G. Assume µ ∈ S ( G ) is a fixed positive measure,that is µ ( g ) > g ∈ G. Then we define the coefficients p fg,h , where f, g, h ∈ G in the following way: p fg,h = µ ( g ) µ ( f + g + H ) , if h ∈ f + g + H ;0 otherwise . It is easy to check that for arbitrary f, g, h ∈ G the conditions (3) aresatisfied. It is also evident that if H = { e } , where e is the neutral lement of group G, then p fg,h = h = f + g ;0 , otherwiseand if H = G, then p fg,h = µ ( h ) ∀ f, g ∈ G . In the common case q.s.o. V on S ( G ) is defined as( V x ) h = X f,g ∈ G p fg,h x f x g for all h ∈ G, where x = { x t , t ∈ G } ∈ S ( G ) and p fg,h as above.If H = { e } then the q.s.o. V is defined as( V x ) h = X f,g ∈ G,f + g = h x f x g and if H = G, the q.s.o. is defined as( V x ) h = µ ( h )for arbitrary h ∈ G. Let us fix a positive measure µ ∈ S ( G ) and subgroup H of group G. Assume µ H is the factor-measure on factorgroup G/H, that is µ H ( g + H ) = X h ∈ H µ ( g + h ) or any g ∈ G and V H is a q.s.o. on S ( G/H ) , which is defined bymeasure µ H . It is easy to show, that the trajectorial behaviour of V and V H are similar,and so it is enough to study the q.s.o. generatedby the trivial subgroup.Below we consider q.s.o. constructed by the trivial subgroup H = { e } . Let ν ∈ S ( G ) be a Haar measure on G . As G is a finite Abeliangroup,the Haar measure ν on G is a uniform distribution on G ,that isit is a centre of the simplex S ( G ). We prove the following Theorem 2
Almost all orbits tend to the center of the simplex.Proof:
The following lemma is a key ingredient:
Lemma 1 || V x || ∞ ≤ || x || ∞ . Proof of lemma:
Let’s define the following function f ( p ) = f n ( p ) = max P n x i =1 ,x i ∈ [0 ,p ] n X x i . Notice that there is no dependence on n. Moreover, f has an explicitform: f ( p ) = kp + (1 − kp ) if p ∈ [ 1 k + 1 , k ] . e can prove this formula by induction .The sum P x i cannot have a maximum in the interior of the do-main (by the Lagrange method). So, we can look for the maximumat the set x n = p. It follows that f n ( p ) = p + max x i ∈ [0 ,p ] , P n − x i =1 − p n − X x i Let x i = (1 − p ) y i .T hen P n − y i = 1 and y i ∈ [0 , p − p ]. So, thefollowing recurrent formula is true: f n ( p ) = p + (1 − p ) f n − ( p − p ) . This allows an easy check of the expected formula. Now it is easy tosee that f ( p ) − p = kp − p +(1 − kp ) = (1 − kp )(1 − kp − p ) = (1 − kp )(1 − ( k +1) p ) ≤ , that is f ( p ) ≤ p. From Cauchy-Bunyakowski inequality:(
V x ) i ≤ sX j ∈ G x j sX j ∈ G x i − j = X j ∈ G x j If max x i ≤ p then, f ( p ) ≤ p implies ( V x ) i ≤ p or max( V x ) i ≤ p. So, the norm || x || ∞ decreases on the orbits. Lemma is thus proven. ctually, the set { x : || x || ∞ ≤ p } is mapped to the set { x : || x || ∞ ≤ f ( p ) } . We can easily check, that the iterations of the function f tendto its fixed points.Now consider a point x. We consider the case, when a point in orbitdoes not tend to a centre. It’s ω − limit set lies at a set { x : g ( x ) = k } . This ω − limit set is finite. So, it is nothing else, then a periodic orbit.We consider the periodic orbits on the set || x || ∞ = k . Since || V x || ∞ = k , there exists k coordinates of x equal to k . All other coordinates are zeros. This is also true for
V x.
So, thereexists sets A = { i : x i = k } and B = { j : ( V x ) j = k } , such that | A | = | B | = k and1. ∀ i ∈ B | ( i − A ) T A | = k → i − A = A, ∀ i / ∈ B | ( i − A ) T A | = 0 → ( i − A ) T A = ∅ . From the second item: ∀ i / ∈ B ∀ x, y ∈ A : i − x = y ; ∀ i / ∈ B ∀ j ∈ A + A, i = j ; A + A ⊂ B. rom the first item ∀ i ∈ B ∃ j, k ∈ A : i = j + k ; B ⊂ A + A. So, there exists a set A, such that | A | = | A + A | = k, ∀ i ∈ A + A : i − A = A. Claim:
This is equivalent to the following property: ∀ i, j, k ∈ A : i + j − k ∈ A. Proof: → ∀ x, y ∈ A x + y − A = A. ∀ x, y, z ∈ A x + y − z ∈ A. ← ∀ x, y ∈ A x + y − A ⊂ A. ∀ z ∈ A + A z − A ⊂ A. Since | z − A | = | A | , z − A = A. ∀ a, i, j ∈ A a + i − j ∈ A. ∀ i, j ∈ A A + i − j ⊂ A. ∀ i, j ∈ A A + i ⊂ A + j. So, ∀ i, j ∈ A A + i = A + j, and hence , A + A = [ j ∈ A ( A + j ) = A + i. So, | A + A | = k. The claim is proved.This property is equivalent to the following : there exists a point p ∈ A and a subgroup H, such that A = p + H. So, the points x, V x, V x, · · · correspond to the sets 2 p + H, p + H, p + H, · · · . Actually, such sequences are pre-periodic.We now prove the instability of such periodic orbits. Let l be it’speriod and V l = T be the corresponding first return map. Then T has an instability direction (an eigenvector, whose eigenvalue is greaterthan 1). It is easy to check, that vector e with coordinates e s = , if s ∈ i + H ; − if s ∈ j + H ;0 , other casesbelongs to the plane { x : P x i = 0 } (a tangent plane for the simplex)and realizes this instability direction.There is no such directions for the centre by the coincidense ofeach two classes.So, the basin of attraction for such orbit consists of subvarieties ofstrictly positive codimension.We conclude that almost all orbits tendto the centre of the simplex. Remark
We can also consider an infinite-dimensional case. All precedingarguments remain true. But there is no natural measure on a sim- lex,and so we cannot use the term ”almost all”.Another disappointment is that ”most” of the orbits diverge inthe l topology. It follows from the fact, that for ”most” orbits thesequence || V n x || ∞ tends to zero. All possible l limit points are l ∞ limit points. Since the only l ∞ limit point is zero, then the subsequenceof the orbit l tends to a point, which does not belong to a simplex.This contradiction implies the absense of l limit points. Acknowledgements
This research was supported in part by the Uz.R. grant F-2.1.56.
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