DYNAMICS OF NEWTON’S FUNCTIONS OF BARNA’S POLYNOMIALS

We define Barna's polynomials as real polynomials with all real 
roots of which at least four are distinct. In this paper, we 
study the dynamics of Newton's functions of such polynomials. We 
also give the upper and lower bounds of the Hausdorff dimension 
of exceptional sets of these Newton's functions.


Introduction.
Newton's method is a well-known iterative method used to locate the roots of functions.Barna, [1,2,3,4], proved that for a real polynomial P (x) with only simple real roots of which at least four are distinct, the exceptional set of initial points of its Newton's function N(x) (the set of x ∈ R such that N j (x) does not converge to any root of P , where N j (x) denotes the jth iterate of N) is homeomorphic to a Cantor subset of R which has the Lebesgue measure zero.Wong [7], generalized this result to real polynomials having all real roots (not necessarily simple) of which at least four are distinct (which will be called Barna's polynomials) by using a symbolic dynamics approach.In this paper, we will investigate the symbolic dynamics of Newton's functions of Barna's polynomials.Furthermore, we give the upper and lower bounds of the Hausdorff dimension of the exceptional sets.

Symbolic dynamics of Newton's functions
Definition 2.1.A real polynomial with all real roots of which at least four are distinct is called a Barna's polynomial.Thus P (x) is a Barna's polynomial if and only if where c is a nonzero real constant, Definition 2.2.The Newton's function N f (x) of a function f (x) is defined as where f (x) is the derivative of f (x).
Let P (x) = c n i=1 (x − r i ) m i be a Barna's polynomial and N P (x) be the Newton's function of P .The following are well-known properties of N P (see [1,2,3,4,7]).
(a) For each i = 1, 2,...,n − 1, there exists c i ∈ (r i ,r i+1 ) such that c i is a zero of P (x), the derivative of P (x), and {c 1 ,c 2 ,...,c n−1 } is exactly the set of all zeros of P which are not zeros of P .
(b) In each of the intervals (−∞,c (e) lim k→∞ N k P (x) = r 1 for all x ∈ (−∞,c 1 ) and lim k→∞ N k P (x) = r n for all x ∈ (c n , +∞).(f) For each i = 2, 3,...,n − 1, the interval (c i−1 ,c i ) contains exactly one period-two cycle of N P , say, at {α i ,β i } where Definition 2.3.Let P (x) and N P (x) be as above.The exceptional set Λ of N P is defined as the complement of the set of real numbers x such that Remark 2.4.Note that Λ consists of points where N k P is well defined for each k ∈ N and never converge to any r i .
Since our main interest is on the set Λ, those points which are not in Λ together with their preimages will be removed from R. From this we have the following result on period-two cycle of Newton's function.

Proposition 2.5. The function N P has a period-two cycle at {α, β} such that
Proof.Since (−∞,c 1 ) and (c n−1 , ∞) are not in Λ, we remove these sets together with their preimages.Let . Then (y 0 , +∞) Λ and we remove this interval.Next let Then (−∞,z 1 ) Λ and we remove this interval.Applying this procedure repeatedly we get two sequences of points {y i } ∞ i=1 and {z j } ∞ j=1 where (2.3) Thus lim i→∞ y i = β and lim j→∞ z j = α exist.As a result, Similarly, we have This completes the proof of the proposition.
For each i = 2, 3,...,n − 1, we have by I 1 ,I 2 ,...,I 2n−4 , respectively, and let I = 2n−4 i=1 I i .With a similar approach used by Wong [7], we shall define the transition matrix V associated to N P .This matrix V will determine the symbolic dynamics of N P .Let V = (v ij ) be a (2n − 4) × (2n − 4) matrix of zeros and ones defined by for i, j ∈ {1, 2,...,2n − 4}.From this definition and properties of N P , it is easily seen that V is a (2n − 4) × (2n − 4) matrix built from the following 2 × 2 matrices: In fact V can be interpreted as an (n − 2) × (n − 2) matrix of matrices as follows: (1) With the same technique in [7], V is irreducible and we can show that N P restricted to Λ is conjugate to the one-sided shift map σ on the set Σ V 2n−4 where is the symbolic sequences space consisting of 2n − 4 symbols (cf.[6]).
Remark 2.6.From [6], we have card(Per k σ ) = Tr(V k ), where card(Per k σ ) denotes the number of points of period k of the shift map σ and Tr(V k ) is the trace of V k .Remark 2.7.By MATHEMATICA, we compute that Tr( where V is the transition matrix associated to Newton's function of a Barna's polynomial with n distinct real roots.
We summarize this section as follows.
Theorem 2.8.Let P (x) = c n i=1 (x − r i ) m i be a Barna's function and N P the Newton's function of P .Let Λ be the exceptional set of N P .Then Λ is a Cantor subset of R and N P restricted to Λ is conjugate to the one-sided shift map on Σ V 2n−4 .
Remark 2.9.There is some difference between our proof of Theorem 2.8 and the proof of a similar result by Wong in [7].In our proof we use the fact that the exceptional set Λ lies between the period-two cycle {α, β} as stated in Proposition 2.5 and hence we can explicitly define the transition matrix V .

Hausdorff dimension of exceptional sets.
Let Λ be the exceptional set of Newton's function of a Barna's polynomial with n distinct real roots.In this section, we give the upper and lower bounds of the Hausdorff dimension of Λ.The technique we will use here is similar to the one used in [5].We first note that N −1 Let the interval I be the same as in the previous section.Then I has n − 2 preimages under N P each in the interval where We now state and prove the result on the estimation of the Hausdorff dimension of Λ.
Proof.For each k ≥ 1, let S k and L k be the lengths of the smallest and largest intervals in Λ k , respectively.For each k ≥ 0, we get Since Λ is compact, any covering {U i } of Λ can be refined to a finite cover, where each element of this cover contains exactly one for some sufficiently large p.Then we get If α < ln(n−2)/ ln M k , then this diverges as p → ∞, that is, as the covering gets smaller.Thus dim Λ ≥ ln(n − 2)/ ln M k .By letting k → ∞, we have dim Λ ≥ ln(n − 2)/ ln M. Similarly, for a given > 0 and for some sufficiently large p, there exists a covering of Λ such that each element of the cover contains exactly one interval x −r i ) m i be a Barna's polynomial and let N P (x) be its Newton's function.Let M(x) = kN P (x/k), k is a nonzero real constant.Then M is the Newton's function of Barna's polynomial of the form Q(x) = c 0 (x −1) m 1 n−1 i=1 (x − r i /r n ) m i and M is conjugate to N P via the map h(x) = kx.As a result, α is a periodic point of N P if and only if kα is a periodic point of M and N P (α) is equal to M (kα).Consequently, dynamics of M and N P on their exceptional sets are the same and the Hausdorff dimensions of their exceptional sets are equal.As a result, it suffices to consider the dynamics of Newton's functions of Barna's polynomials which have 1 as the largest root.