Orthogonal polynomials on generalized Julia sets
aa r X i v : . [ m a t h . D S ] J un ORTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS
G ¨OKALP ALPAN AND ALEXANDER GONCHAROVA
BSTRACT . We extend results by Barnsley et al. about orthogonal polynomials on Julia setsto the case of generalized Julia sets. The equilibrium measure is considered. In addition, wediscuss optimal smoothness of Green’s functions and Parreau-Widom criterion for a specialfamily of real generalized Julia sets.
1. I
NTRODUCTION
Let f be a rational function in C . Then the set of all points z ∈ C such that the sequenceof iterates ( f n ( z )) ¥ n = is normal in the sense of Montel is called the Fatou set of f . Thecomplement of the Fatou set is called the Julia set of f and we denote it by J ( f ) . We use theadjective autonomous in order to refer to these usual Julia sets in the text.Potential theoretical tools for Julia sets of polynomials were developed in [8] by HansBrolin. Orthogonal polynomials for polynomial Julia sets were considered in [4, 5]. Barnsleyet al. show how one can find recurrence coefficients when the Julia set J ( f ) corresponding toa nonlinear polynomial is real. Ma˜n´e and Rocha, in [22], show that Julia sets are uniformlyperfect in the sense of Pommerenke and in particular they are regular with respect to theDirichlet problem.Let ( f n ) be a sequence of rational functions. Define F ( z ) : = z and F n ( z ) = f n ◦ F n − ( z ) forall n ∈ N , recursively. The union of the points z such that the sequence ( F n ( z )) ¥ n = is normalis called the Fatou set for ( f n ) and the complement of the Fatou set is called the Julia setfor ( f n ) . We use the notation J ( f n ) to denote it. These sets were introduced in [15]. For ageneral overview we refer the reader to the paper [10]. For a recent discussion of Chebyshevpolynomials on these sets, see [1].In this paper, we consider orthogonal polynomials with respect to the equilibrium measureof J ( f n ) where ( f n ) is a sequence of nonlinear polynomials satisfying some mild conditions.To our knowledge, this paper is the first attempt dealing with the orthogonal polynomialsin this generality although considerable work (see e.g. [4, 5, 6]) has been done for theautonomous case and there are some results (see e.g. [2, 26]) concerning the orthogonalpolynomials on sets constructed using compositions of infinitely many polynomials. Whilethe focus of [26] is quite different than what we discuss, a particular family of sets consideredin [2, 19] clearly presents generalized Julia sets.In Section 2, we give background information about the properties of J ( f n ) regarding po-tential theory. In Section 3, we prove that for certain degrees, orthogonal polynomials asso-ciated with the equilibrium measure of J ( f n ) are given explicitly in terms of the compositions F n . In Section 4, we show that the recurrence coefficients can be calculated provided that Date : Received: date / Accepted: date.2010
Mathematics Subject Classification.
Key words and phrases.
Julia sets, Parreau-Widom sets, orthogonal polynomials, Jacobi matrices.The authors are partially supported by a grant from T¨ubitak: 115F199. J ( f n ) is real. These two results generalize Theorem 3 in [4] and Theorem 1 in [5] respec-tively. In addition to these results we discuss resolvent functions and a general method toconstruct real Julia sets. Techniques that we use here are rather different compared to thoseof autonomous setting. This is mostly due to the fact that, in the generalized case, Julia setsdo not have complete invariance but we only have the properties given in part ( e ) of Theorem1. In Section 6, we consider a quadratic family of polynomials ( f n ) such that the set K ( g ) = J ( f n ) is a modification of the set K ( g ) from [19]. In terms of the parameter g we give a crite-rion for the Green function G C \ K ( g ) to be optimally smooth. In the last section, a criterionis presented for K ( g ) to be a Parreau-Widom set.2. P RELIMINARIES
Polynomial Julia sets are one of the most studied objects in one dimensional complexdynamics. For classical results related to potential theory, see [8]. For a more general expo-sition we refer to the monograph [25] and the survey [21].In this paper, we study in the more general framework of Julia sets. Clearly, Theorem 3.3and Theorem 4.1 are also valid for the autonomous Julia sets.Let the polynomials f n ( z ) = (cid:229) d n j = a n , j · z j be given where d n ≥ a n , d n = n ∈ N .Following [10], we say that ( f n ) is a regular polynomial sequence if for some positive realnumbers A , A , A , the following properties are satisfied: • | a n , d n | ≥ A , ∀ n ∈ N . • | a n , j | ≤ A | a n , d n | for j = , , . . . , d n − n ∈ N . • log | a n , d n | ≤ A · d n for all n ∈ N .We use the notation ( f n ) ∈ R if ( f n ) is a regular polynomial sequence. We remark that,for a sequence ( f n ) ∈ R , the degrees of polynomials need not to be the same and they donot have to be bounded above either. Julia sets J ( f n ) when ( f n ) ∈ R were introduced andconsidered in [11] and all results given in the next theorem are from Section 2 and Section 4of the paper [10]. While (2.1) is contained in the proof of Theorem 4.2 in [10], (2.2) followsby comparing the right parts of these two equations, using that G C \ J ( fn ) has a logarithmicsingularity at infinity and F k ( z ) goes locally uniformly to ¥ for such z . Theorem 2.1.
Let ( f n ) ∈ R . Then the following propositions hold:(a) The set A ( f n ) ( ¥ ) : = { z ∈ C : F k ( z ) goes locally uniformly to ¥ } is an open connectedset containing ¥ . Moreover, for every R > satisfying the inequalityA R (cid:18) − A R − (cid:19) > , the compositions F n ( z ) goes locally uniformly to infinity whenever z ∈ △ R where △ R = { z ∈ C : | z | > R } . (b) A ( f n ) ( ¥ ) = ∪ ¥ k = F k − ( △ R ) and f n ( △ R ) ⊂ △ R if R > satisfies the inequality givenin part ( a ) . Furthermore, we have J ( f n ) = ¶ A ( f n ) ( ¥ ) . RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 3 (c) J ( f n ) is regular with respect to the Dirichlet problem. The Green function for thecomplement of the set is given by (2.1) G C \ J ( fn ) ( z ) = (cid:26) lim k → ¥ d ··· d k log | F k ( z ) | if z ∈ A ( f n ) ( ¥ ) , otherwise.Moreover, (2.2) G C \ J ( fn ) ( z ) = lim k → ¥ d · · · d k G C \ J ( fn ) ( F k ( z )) where z ∈ A ( f n ) ( ¥ ) . In both (2.1) and (2.2) , limits hold locally uniformly in A ( f n ) ( ¥ ) . (d) The logarithmic capacity of the compact set J ( f n ) is given by the expression Cap ( J ( f n ) ) = exp − lim k → ¥ k (cid:229) j = log | a j , d j | d · · · d j ! . (e) F − k ( F k ( J ( f n ) )) = J ( f n ) and J ( f n ) = F − k ( J ( f k + n ) ) for all k ∈ N . Here we use the notation ( f k + n ) = ( f k + , f k + , f k + , . . . ) . We have to note that for the sequences ( f n ) ∈ R satisfying the additional condition d n = d for some d ≥
2, there is a nice theory concerning topological properties of Julia sets. Fordetails, see [13, 23].Before going any further, we want to mention the results from [4] and [5] concerningorthogonal polynomials for the autonomous Julia sets. Let f ( z ) = z n + k z n − + . . . + k n be a nonlinear monic polynomial of degree n and let P j denote the j -th monic orthogonalpolynomial associated to the equilibrium measure of J ( f ) . Then we have,(a) P ( z ) = z + k / n .(b) P ln ( z ) = P l ( f ( z )) , for l = , , . . . (c) P n l ( z ) = f l ( z ) + k / n for l = , , . . . , where f l is the l -th iteration of the function f .In Theorem 3.3, we recover parts ( a ) and ( c ) in a more general setting. Even withouthaving the analogous equations to part ( b ) , recurrence coefficients appear as the outcome ofTheorem 4.1.Throughout the whole article when we say that ( f n ) ∈ R then the sequences ( d n ) , ( a n , j ) , ( A i ) i = will be used just as in the definition given in the beginning of this section and F n ( z ) will stand for f n ◦ . . . ◦ f ( z ) . Thus F n is a polynomial with the leading coefficient ( a , d ) d ··· d l ( a , d ) d ··· d l · · · a l , d l of degree d · · · d n . For a compact non-polar set K , we denotethe Green function of W with pole at infinity by G C \ K where W is the connected componentof C \ K containing ¥ . We use m K to denote the equilibrium measure of K . Convergence ofmeasures is considered in weak-star topology. In addition, we consider and count multipleroots of a polynomial separately.3. O RTHOGONAL POLYNOMIALS
We begin with a lemma due to Brolin [8].
Lemma 3.1.
Let K and L be two non-polar compact subsets of C such that K ⊂ L. Let ( m n ) ¥ n = be a sequence of probability measures supported on L that converges to a measure m supported on K. Suppose that the following two conditions hold where U n ( z ) stands forthe logarithmic potential for the measure m n and V K is the Robin constant for K: G ¨OKALP ALPAN AND ALEXANDER GONCHAROV (a) lim inf n → ¥ U n ( z ) ≥ V K on K.(b) supp ( m K ) = K.Then m = m K . Let ( f n ) ∈ R . Then, by the fundamental theorem of algebra (FTA), F k ( z ) − a = d · · · d k solutions counting multiplicities. For a fiven k , let us define the normalized countingmeasure as n ak = d ··· d k (cid:229) d ··· d k l = d z l where z , . . . , z d ··· d k are the roots of F k ( z ) − a . In [8] andlater on in [9], it is shown that n ak → m J ( fn ) for a proper a where in the first article f n = f witha monic nonlinear polynomial f and in the second one f n ( z ) = z + c n . Our technique usedbelow is the same in essence with the proofs in [8, 9]. Due to some minor changes and forthe convenience of the reader, we include the proof of the theorem. Theorem 3.2.
Let ( f n ) ∈ R . Then for a ∈ C \ D satisfying the condition (3.1) | a | A (cid:18) − A | a | − (cid:19) > , we have n ak → m J ( fn ) .Proof. Choose a number a ∈ C \ D satisfying (3.1). Let K : = J ( f n ) and L : = { z ∈ C : | z | ≤ | a |} .Then, by part ( b ) of Theorem 2.1, K ( L . Moreover, since K is regular with respect to theDirichlet problem and K is equal to the boundary of the component of C \ K that contains ¥ ,we have (see e.g. Theorem 4.2.3. of [27]) that supp ( m K ) = K .Observe that, F k − ( a ) ∩ A ( f n ) ( ¥ ) is contained in L for all k ∈ N by part ( b ) of Theorem2.1. Thus, ( n ak ) ¥ k = has a convergent subsequence ( n ak l ) ¥ l = by Helly’s selection principle (seee.g. Theorem 0.1.3. in [29]). Let us denote the limit by m . The set ∪ F k − ( a ) can notaccumulate to a point z in A ( f n ) ( ¥ ) , since this would contradict with the fact that F k ( z ) goeslocally uniformly to ¥ by part ( a ) of Theorem 2.1. Thus, supp ( m ) ⊂ ¶ A ( f n ) ( ¥ ) = K .Now, we want to show that lim inf l → ¥ U k l ( z ) ≥ V K for all z ∈ K . Let z ∈ K where U k denotethe logarithmic potential for n ak . We have | F k l ( z ) − a | = | ( a , d ) d ··· d kl || ( a , d ) d ··· d kl | · · · | a k l , d kl | d ··· d kl (cid:213) j = | z − z j , k l | , for some z j , k l ∈ L . Thus,(3.2) U k l ( z ) = (cid:229) d ··· d kl j = log | z − z j , k l |− d · · · d k l = d ··· d kl (cid:229) j = log | a j , d j | d · · · d j − log | F k l ( z ) − a | d · · · d k l . Using part ( d ) of Theorem 2.1 and the fact that | F k ( z ) | ≤ | a | for z ∈ K , we see that thefollowing inequality follows from (3.2):lim inf l → ¥ U k l ( z ) ≥ lim inf l → ¥ d ··· d kl (cid:229) j = log | a j , d j | d · · · d j − log | a | d · · · d k l ≥ V K . Hence, by Lemma 3.1, we have n ak l → m K . Since ( n ak l ) is an arbitrary convergent subsequence, n ak → m K also holds. (cid:3) RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 5
In the next theorem, we use algebraic properties of polynomials as well as analytic prop-erties of the corresponding Julia sets. Let f ( z ) = a n z n + a n − z n − + . . . a be a nonlinearpolynomial of degree n and let z , z , . . . , z n be roots of f counting multiplicities. Then, for k = , , . . . n − s k ( f ( z )) + a n − a n s k − ( f ( z )) + . . . + a n − k + a n s ( f ( z )) = − k a n − k a n , where s k ( f ( z )) : = (cid:229) nj = ( z j ) k .For the proof of (3.3), see [24] among others. Note that, none of these equations includethe term a . This implies that the values ( s k ) n − k = are invariant under translation of the function f , i.e.(3.4) s k ( f ( z )) = s k ( f ( z ) + c ) for any c ∈ C . Let ( P j ) ¥ j = denote the sequence of monic orthogonal polynomials associatedto m J ( fn ) where deg P j = j . Now we are ready to prove our first main result. Theorem 3.3.
For ( f n ) ∈ R , we have the following identities:(a) P ( z ) = z + d a , d − a , d . (b) P d ··· d l ( z ) = ( a , d ) d ··· d l ( a , d ) d ··· d l · · · a l , d l (cid:18) F l ( z ) + d l + a l + , d l + − a l + , d l + (cid:19) . Proof. (a) Let ( f n ) ∈ R be given and a ∈ C \ D satisfy (3.1). Fix an integer m greater than 1.By FTA, The solutions of the equation F m ( z ) = a satisfy an equation of the form (cid:0) F m − ( z ) − b m − (cid:1) . . . ( F m − ( z ) − b d m m − ) = , where b m − , . . . , b d m m − ∈ C . The d · · · d m − roots of the equation F m − − b jm − = ( F m − ( z ) − b , jm − ) . . . ( F m − ( z ) − b d m − , jm − ) = , with some b , jm − , . . . , b d m − , jm − . Continuing this way, the points satisfying the equation F m ( z ) = a can be grouped into d · · · d m parts of size d such that each part consists of the roots of anequation f ( z ) − b j = , for j ∈ { , . . . , d · · · d m } and b j ∈ C . If for each j , we denote the normalized countingmeasure on the roots of f ( z ) − b j by l j , then n am = d · · · d m d ··· d m (cid:229) j = l j . Hence, by (3.3) and (3.4), Z z d n am = d · · · d m d ··· d m (cid:229) j = Z z d l j = d · · · d m d ··· d m (cid:229) j = s ( f ( z ) − b j ) d = d · · · d m d ··· d m (cid:229) j = s ( f ( z )) = − d a , d − a , d . G ¨OKALP ALPAN AND ALEXANDER GONCHAROV
Since n am converges to the equilibrium measure of J ( f n ) by Theorem 3.2, the result follows.(b) Let m , l ∈ N where m > l +
1. As above, the roots of the equation F m ( z ) = a where a ∈ C \ D satisfies (3.1), can be grouped into d l + · · · d m parts of size d · · · d l + such that eachpart obeys an equation of the form F l + ( z ) − b jl + = , for j = , , . . . , d l + · · · d m . Recall that F l + ( z ) = f l + ( t ) with t = F l ( z ) . By FTA, we have f l + ( t ) − b jl + = ( t − b , jl ) · · · ( t − b d l + , jl ) for some b , jl , . . . , b d l + , jl . By(3.3) and (3.4), for k ∈ { , . . . , d l + − } and j , j ′ ∈ { , . . . , d l + · · · d m } , we have s k ( f l + ( t ) − b jl + ) : = d l + (cid:229) r = ( b r , jl ) k = d l + (cid:229) r = ( b r , j ′ l ) k = s k ( f l + ( t ) − b j ′ l + ) . Now we can rewrite F l + ( z ) − b jl + = ( F l ( z ) − b , jl ) · · · ( F l ( z ) − b d l + , jl ) = j asabove. Let us denote the normalized counting measures on the roots of F l ( z ) − b r , jl = l r , j for r = , . . . , d l + and j = , . . . , d l + · · · d m . Clearly, this yields(3.5) n am = d l + · · · d m d l + ··· d m (cid:229) j = d l + d l + (cid:229) r = l r , j = d l + · · · d m d l + ··· d m (cid:229) j = d l + (cid:229) r = l r , j . Thus, by using (3.5), (3.3) and (3.4), we deduce that Z F l ( z ) d n am = d l + · · · d m d l + ··· d m (cid:229) j = d l + (cid:229) r = Z F l ( z ) d l r , j = d l + · · · d m d l + ··· d m (cid:229) j = d l + (cid:229) r = b r , jl = d l + · · · d m d l + ··· d m (cid:229) j = s ( f l + ( t ) − b jl + )= d l + · · · d m d l + ··· d m (cid:229) j = s ( f l + ( t ))= − d l + a l + , d l + − a l + , d l + . To shorten notation, we write c instead of d l + a l + , dl + − a l + , dl + . Thus, we have(3.6) Z ( F l ( z ) + c ) d n am = . Let us show that the integrand is orthogonal to z k with 1 ≤ k ≤ d · · · d l − l r , j , as above, we have Z ( F l ( z ) + c ) z k d l r , j = d · · · d l (cid:16) b r , jl + c (cid:17) · s k (cid:16) F l ( z ) − b r , jl (cid:17) . RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 7
By (3.4), s k (cid:16) F l ( z ) − b r , jl (cid:17) = s k ( F l ( z )) , so it does not depend on r or j . This and the repre-sentation (3.5) imply that Z ( F l ( z ) + c ) z k d n am = d l + · · · d m d l + ··· d m (cid:229) j = d l + (cid:229) r = Z ( F l ( z ) + c ) z k d l r , j = s k ( F l ( z )) d . . . d l Z ( F l ( z ) + c ) d n am , where the last term is equal to 0, by (3.6). It follows that ( F l ( z ) + c ) ⊥ z k for k ≤ deg F l − L ( m J ( fn ) ) , since n am converges to the equilibrium measure of J ( f n ) . This completes the proofof the theorem. (cid:3)
4. M
OMENTS AND RESOLVENT FUNCTIONS
In this section we consider Julia sets that are subsets of the real line.If m is a probability measure which has infinite compact support in R , then the monicorthogonal polynomials ( P n ) ¥ n = satisfy a recurrence relation P n + ( x ) = ( x − b n + ) P n ( x ) − a n P n − ( x ) , assuming that P = P − =
0. If the moments c n = R x n d m are known for all n ∈ N then we have the formula(4.1) p n ( x ) = √ D n D n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c c . . . c n c c . . . c n + ... ... ... c n − c n . . . c n − x . . . x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where p n is the n -th orthonormal polynomial and D n is the determinant for the matrix M n withthe entries ( M n ) i , j = c i + j for i , j = , , . . . n . From (4.1), one can also calculate recurrencecoefficients ( a n , b n ) ¥ n = . See [35] for a detailed description of the orthogonal polynomials onthe real line. In the next theorem, we show that the moments for the equilibrium measure of J ( f n ) can be calculated recursively whenever ( f n ) ∈ R . Note that c = Theorem 4.1.
Let ( f n ) ∈ R and l > be an integer. Furthermore, letF l ( z ) p l = z d ··· d l + a d d ··· d l − z d d ··· d l − + . . . + a z + a , where p l is the leading coefficient for F l . Then, each moment c k = R x k d m J ( fn ) for k ∈{ , . . . , ( d d · · · d l ) − } is equal to s k ( F l ( z )) d ··· d l where s k ( F l ( z )) can be calculated recursivelyby Newton’s identities.Proof. Let m be an integer greater than l . Consider the roots of the equation F m ( z ) = a where a ∈ △ satisfies the condition (3.1). Then, following the proof of Theorem 3.3, we G ¨OKALP ALPAN AND ALEXANDER GONCHAROV can divide these roots into d l + · · · d m parts of size d · · · d l such that the nodes in each of thegroups constitute the roots of an equation of the form F l ( z ) − b j = , for j = , , . . . , d l + · · · d m . If for each j we denote the normalized counting measure on theroots of F l ( z ) − b j by l j , then by (3.3) and (3.4), this leads to Z x k d n am = d l + · · · d m d l + ··· d m (cid:229) j = Z x k d l j = d l + · · · d m d l + ··· d m (cid:229) j = s k ( F l ( z ) − b j ) d · · · d l = d l + · · · d m d l + ··· d m (cid:229) j = s k ( F l ( z )) d · · · d l = s k ( F l ( z )) d · · · d l , for k = , . . . , ( d d · · · d l ) −
1. Since the weak star limit of the sequence ( d n am ) is theequilibrium measure of the Julia set by Theorem 3.2, we have R x k d m J ( fn ) = s k ( F l ( z )) d ... d l whichconcludes the proof. (cid:3) In Sections 3-5 of [2], orthogonal polynomials and recurrence coefficients are discussedfor the quadratic case. It would be interesting to obtain similar results for m J ( fn ) if we onlyassume that ( f n ) ∈ R and J ( f n ) ⊂ R .For two bounded sequences ( a n ) ¥ n = and ( b n ) ¥ n = with a n > b n ∈ R for n ∈ N , the as-sociated (half-line) Jacobi operator H : ℓ ( N ) → ℓ ( N ) is given by ( Hu ) n = a n u n + + b n u n + a n − u n − for u ∈ ℓ ( N ) and a : =
0. Here, ℓ ( N ) denotes the space of square summablesequences in N . The spectral measure of H for the cyclic vector d = ( , , , . . . ) T is just theone which has a n , b n ( n = , . . . ) as the recurrence coefficients.Let J ( f n ) ⊂ [ − M , M ] for some M ∈ R where ( f n ) ∈ R . If we denote the Jacobi operatorassociated with m J ( fn ) by H ( f n ) then the resolvent function R ( f n ) is defined as R ( f n ) ( z ) : = Z d m J ( fn ) ( x ) x − z = h ( H ( f n ) − z ) − d , d i for z ∈ C \ J ( f n ) . Note that R ( f n ) is an analytic function. If f n = f for a nonlinear polynomial f for all n ∈ N then the resolvent function satisfies a functional equation:(4.2) R ( f ) ( z ) = f ′ ( z ) deg f R ( f ) ( f ( z )) . See e.g. [6] for a discussion of resolvent functions and operators associated with theequilibrium measure of autonomous polynomial Julia sets. It is well known that (see e.g. p.53 in [31]) for z ∈ C \ D M ( ) (4.3) R ( f n ) ( z ) = − ¥ (cid:229) n = c n z − ( n + ) where c n is the n -th moment for m J ( fn ) , D M ( ) is the open ball centered at 0 with radius M in C and the series at (4.3) is absolutely convergent in the corresponding domain. RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 9
We define the ¶ operator as ¶ = ¶ x − i ¶ y . If g is a harmonic function on a simply connected domain D ⊂ C then (see e.g. Theorem1.1.2 in [27]) there is an analytic function h on D such that g = Re h holds. Moreover, wehave h ′ ( z ) = ¶ g ( z ) . Furthermore, G C \ J ( fn ) ( z ) = log ( Cap ( J ( f n ) ) − ) − U m J ( fn ) ( z ) holds where U m J ( fn ) is the logarithmic potential for m J ( fn ) . In addition, for each z ∈ C \ J ( f n ) ,there is a d > h (which may depend on z ) such that (see e.g. p.87 in [14]) h ′ ( z ) = R ( f n ) ( z ) and Re h = U m J ( fn ) for z ∈ D d ( z ) . By harmonicity of U m J ( fn ) thisimplies(4.4) 2 ¶ G C \ J ( fn ) ( z ) = − ¶ U m J ( fn ) ( z ) = − R ( f n ) ( z ) for all z ∈ C \ J ( f n ) . The next theorem follows from the discussion above. Theorem 4.2.
Let J ( f n ) ⊂ R provided that ( f n ) ∈ R . Then the following functional equationholds where the limit exists locally uniformly in C \ J ( f n ) : (4.5) R ( f n ) ( z ) = lim k → ¥ R ( f n ) ( F k ( z )) F ′ k ( z ) d · · · d k . Proof.
If we apply ¶ to both sides of (2.2), it is permitted to change the differentiation andlimit since (see e.g. p. 16 in [3]) G C \ J ( fn ) is harmonic in A ( f n ) ( ¥ ) \ ¥ . Note that A ( f n ) ( ¥ ) \ ¥ = C \ J ( f n ) here since J ( f n ) lies on R . Hence, we have(4.6) ¶ G C \ J ( fn ) ( z ) = lim k → ¥ ¶ G C \ J ( fn ) ( F k ( z )) F ′ k ( z ) d · · · d k where the limit on the right side of (4.6) holds locally uniformly. Using (4.4) and (4.6), wehave (4.5) immediately. (cid:3) Remark 4.3.
Provided that f n = f for a fixed nonlinear polynomial f in Theorem 4.2, (4.5)reduces to (4.2) if we put f ( z ) instead of z in both sides of (2.2) and follow the steps of theproof of Theorem 4.2. 5. C ONSTRUCTION OF REAL J ULIA SETS
Let f be a nonlinear real polynomial with real and simple zeros x < x < . . . < x n anddistinct extremas y < . . . < y n − with | f ( y i ) | > i = , , . . . , n −
1. Then we say that f is an admissible polynomial. Note that in the literature the last condition is usually given as | f ( y i ) | ≥
1. We list useful features of preimages of admissible polynomials.
Theorem 5.1. [16]
Let f be an admissible polynomial of degree n. Thenf − ([ − , ]) = ∪ ni = E i where E i is a closed non-degenerate interval containing exactly one root x i of f for each i.These intervals are pairwise disjoint and m f − ([ − , ]) ( E i ) = / n. We say that an admissible polynomial f satisfies the property ( A ) if (a) f − ([ − , ]) ⊂ [ − , ] , (b) f ( {− , } ) ⊂ {− , } ,(c) f ( a ) = f ( − a ) = ( c ) implies that f is even or odd. Lemma 5.2.
Let g and g be admissible polynomials satisfying ( A ) . Then g : = g ◦ g isalso an admissible polynomial that satisfies ( A ) . Proof.
Let deg g k = n k . Moreover, let ( x j , ) n j = , ( x j , ) n j = be the zeros and ( y j , ) n − j = and ( y j , ) n − j = be the critical points of g , g respectively. Then the equation g ( z ) = g ( z ) = x j , for some j ∈ { , . . . , n } . By ( a ) and ( b ) , the equation g ( z ) = b has n distinct roots for | b | ≤ g ( z ) = b and g ( z ) = b are disjointfor different b , b ∈ [ − , ] . Therefore, g has n n distinct zeros. Similarly, ( g ) ′ ( z ) = g ′ ( g ( z )) g ′ ( z ) = g ′ ( z ) = g ( z ) = y j , for some j ∈ { , . . . , n − } . Theequation g ′ ( z ) = n − ( − , ) . For each of them | g ( z ) | > g ′ ( g ( z )) =
0. On the other hand, for each j ≤ n −
1, the equation g ( z ) = y j , has n distinct solutions with g ′ ( y j , ) =
0. Thus, the total number of solutions for the equation g ′ ( z ) = n − + n ( n − ) = n n − g is admissible. Itis straightforward that for the function g parts ( a ) and ( b ) are satisfied. The part ( c ) is alsosatisfied for g , since arbitrary compositions of even and odd functions are either even orodd. (cid:3) Lemma 5.3.
Let ( f n ) ∈ R be a sequence of admissible polynomials satisfying ( A ) . Then F n is an admissible polynomial with the property ( A ) . Besides, F − n + ([ − , ]) ⊂ F − n ([ − , ]) ⊂ [ − , ] and K = ∩ ¥ n = F − n ([ − , ]) is a Cantor set in [ − , ] .Proof. All statements except the last one follow directly from Lemma 5.2 and the represen-tation F n ( z ) = f n ◦ F n − ( z ) . Let us show that K is totally disconnected.If K is polar then (see e.g. Corollary 3.8.5. of [27]) it is totally disconnected. If K isnon-polar, then (see e.g. Theorem A.16. of [30]), m F − n ([ − , ]) → m K . Suppose that K is nottotally disconnected. Then K contains an interval E such that E ⊂ F − n ([ − , ]) for all n .Since we have m F − n ([ − , ]) ( E ) ≤ / ( d . . . d n ) by Theorem 5.1, convergence of ( m F − n ([ − , ]) ) implies that m K ( E ) =
0. Thus all interior points of E in R are outside of the support of m K .This is impossible by Theorem 4.2.3. of [27] since K = ¶ ( C \ K ) and Cap ( E ) > (cid:3) Here we consider admissible polynomials as polynomials of complex variable.
Lemma 5.4.
Let f be an admissible polynomial satisfying ( A ) . Then | f ( z ) | > + e provided | z | > + e for e > . If | z | = then | f ( z ) | > unless z = ± .Proof. Let deg f = n and x < x < . . . < x n be the zeros of f . By ( c ) , x k = − x n + − k for k ≤ n . In particular, if n is odd, then x ( n + ) / = . Let x i = e >
0. Then, by the law of cosines, the polynomial P x i ( z ) : = z − x i attains minimum of its modulus on the set { z : | z | = + e } at the point z = + e . Therefore | P x i ( z ) | / | P x i ( ± ) | > + e for any z with | z | = + e . Using the symmetry of the roots of f about x =
0, we see that | f ( z ) | = | f ( z ) / | f ( ± ) | > + e for such z .If | z | = | P x i ( z ) | attains its minimum at the points ± . Hence we have | f ( z ) | = | f ( z ) | / | f ( ± ) | > | z | = z = ± (cid:3) RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 11
In the next theorem we use the argument of Theorem 1 in [19].
Theorem 5.5.
Let ( f n ) ∈ R be a sequence of admissible polynomials satisfying ( A ) . ThenK = ∩ ¥ n = F − n ([ − , ]) = J ( f n ) .Proof. Let us prove first the inclusion J ( f n ) ⊂ K . Let R > A R ( − ( A / ( R − ))) >
2. Then by part ( b ) of Theorem 2.1, we have A ( f n ) ( ¥ ) = ∪ ¥ k = F k − ( △ R ) and f n ( △ R ) ⊂ △ R for all n . If we show that | F n ( z ) | > + e for some n ∈ N and for somepositive e , this implies that F n + k ( z ) ∈ △ R for some positive k by Lemma 5.4 and thus z J ( f n ) .Let | z | = + e where e >
0. Then by Lemma 5.4, | F ( z ) | > + e . Hence, z J ( f n ) .Let | z | = z = ±
1. Then using Lemma 5.4, we see that | F ( z ) | >
1. Thus, z J ( f n ) .If we let z ∈ [ − , ] \ K , then there exists a number N ∈ N such that | F N ( z ) | >
1. As aresult, z J ( f n ) .Letting z = x + iy where x K , | y | > | z | < N such that | F N ( x ) | >
1. Since all of the zeros of F n are on the real line by Lemma5.3, we have | F n ( z ) | > | F n ( x ) | >
1. Hence z J ( f n ) .Let z = x + iy where x ∈ K , | y | > | z | <
1. Since K is a Cantor set by Lemma5.3, there exists a number N ∈ N such that n > N implies that each connected componentof F − n ([ − , ]) has length less than y /
8. Let x < x . . . < x d ... d N + be the roots of thepolynomial F N + and E j denote the connected component of F − N + ([ − , ]) containing x j for j = , , . . . , d . . . d N + . Furthermore, let E s = [ a , b ] be the component containing the point x . Observe that | F N + ( a ) | = | F N + ( b ) | =
1. So, in order to show z J ( f n ) , it is enough toshow that | F N + ( z ) | > | F N + ( a ) | .If j < s , then | a − x j | ≤ | x − x j | < | z − x j | .If j = s , then | a − x j | < y / < | y | ≤ | z − x j | .If j > s , then | a − x j | = q | x j − a | ≤ q | x j − x | + | x − a | + | x j − x || x − a | < r | x j − x | + y + y < q | x j − x | + y = | z − x j | . Therefore, | F n ( z ) | >
1. Thus, we have J ( f n ) ⊂ K and C \ K ⊂ A ( f n ) ( ¥ ) .For the inverse inclusion, observe that K ⊂ { z : | F n ( z ) | ≤ n } where { z : | F n ( z ) | ≤ n } ∩ A ( f n ) ( ¥ ) = /0. Since K is contained in the real line and C \ K ⊂ A ( f n ) ( ¥ ) by thefirst part of the proof, we have K ⊂ ¶ A ( f n ) ( ¥ ) = J ( f n ) . (cid:3) Corollary 5.6.
Orthogonal polynomials associated to the equilibrium measure of K and thecorresponding recurrence coefficients (Jacobi coefficients) can be calculated by Theorem 3.3and Theorem 4.1.
6. S
MOOTHNESS OF G REEN ’ S FUNCTIONS
For some generalized Julia sets a deeper analysis can be done. In this section we con-sider a modification K ( g ) of the set K ( g ) from [19] that will quite correspond to Theorem G C \ K ( g ) optimally smooth. Although smoothness properties of Green functions areinteresting in their own rights, in our case the optimal smoothness of G C \ K ( g ) is necessaryfor K ( g ) to be a Parreau-Widom set.Let K ⊂ C be a non-polar compact set. Then G C \ K is said to be H¨older continuous withexponent b if there exists a number A > G C \ K ( z ) ≤ A ( dist ( z , K )) b , holds for all z satisfying dist ( z , K ) ≤ , where dist ( · ) stands for the distance function. Forapplications of smoothness of Green functions, we refer the reader to [7].Smoothness properties of Green functions are examined for a variety of sets. For the com-plement of autonomous Julia sets, see [20] and for the complement of J ( f n ) see [9, 10]. When K is a symmetric Cantor-type set in [ , ] , it is possible to give a sufficient and necessary con-dition in order the Green function for the complement of the Cantor set is H¨older continuouswith the exponent 1 /
2, i.e. optimally smooth. See Chapter 5 in [34] for details.We will use density properties of equilibrium measures. By the next theorem, which isproven in [33], it is possible to associate the density properties of equilibrium measures withthe smoothness properties of Green’s functions.
Theorem 6.1.
Let K ⊂ C be a non-polar compact set which is regular with respect to theDirichlet problem. Let z ∈ ¶ W where W is the unbounded component of C \ K. Then forevery < r < we have r Z m K ( D t ( z )) t dt ≤ sup | z − z | = r G W ( z ) ≤ r Z m K ( D t ( z )) t dt . Let g : = ( g n ) ¥ n = be given such that 0 < g n < / n , e n : = / − g n . Take f n ( z ) = g n ( z − ) + n ∈ N . Thus, F ( z ) = g ( z − ) + F n ( z ) = g n ( F n − ( z ) − ) + n ≥ . It is easy to see that, as a polynomial of real variable, F n is admissible, itsatisfies ( A ) and, in addition, all minimums of F n are the same and equal to 1 − g n . Then K ( g ) = ∩ ¥ n = F − n ([ − , ]) is a stretched version of the set K ( g ) from [19]. Here, G C \ K ( g ) ( z ) = lim n → ¥ − n log | F n ( z ) | . Since the leading coefficient of F n is 2 − n g n g n − · · · g n − , the logarithmic capacity of K ( g ) is 2 exp ( (cid:229) ¥ n = − n log g n ) . If, in addition, for some 0 < c < / g n ≥ c for all n , then ( f n ) ∈ R and, byTheorem 5.5, K ( g ) = J ( f n ) . Without this condition the sequence ( f n ) is not regular, theset K ( g ) is not uniformly perfect (at least if we assume that g n ≤ /
32 for all n ∈ N , seeTheorem 3 in [19]), but polynomials from Theorem 3.3 are still orthogonal, by [2].In the limit case, when all g n = / , F n is the Chebyshev polynomial (of the first kind) T n and K ( g ) = [ − , ] . Let I , : = [ − , ] . The set F − n ([ − , ]) is a disjoint union of 2 n non-degenerate closedintervals I j , n = [ a j , n , b j , n ] with length l j , n for 1 ≤ j ≤ n . We call them basic intervals ofn − th level . The inclusion F − n + ([ − , ]) ⊂ F − n ([ − , ]) implies that I j − , n + ∪ I j , n + ⊂ I j , n RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 13 where a j − , n + = a j , n and b j , n + = b j , n . We denote the gap ( b j − , n + , a j , n + ) by H j , n and the length of the gap by h j , n . Thus, K ( g ) = [ − , ] \ ¥ [ n = [ ≤ j ≤ n H j , n . Let us consider the parameter function v g ( t ) = p − g ( − t ) for | t | ≤ < g ≤ / . This increasing and concave function is an analog of u from [19]. By means of v g we canwrite the endpoints of the basic intervals of n − th level, which are the solutions of F k ( x ) = − ≤ k ≤ n together with the points ± . Namely, F n ( x ) = − F n − ( x ) = ± v g n ( − ) , then F n − ( x ) = ± v g n − ( ± v g n ( − )) , etc. The iterates eventually give 2 n values(6.1) x = ± v g ◦ ( ± v g ◦ ( · · · ± v g n − ◦ ( ± v g n ( − ) · · · ) , which are the endpoints { b j − , n , a j , n } n − j = . The remaining 2 n points can be found similarly,as the solutions of F k ( x ) = − ≤ k < n and ± . As in Lemma 2 in [19], min ≤ j ≤ n l j , n is realized on the first and the last intervals. Sincethe rightmost solution of F n ( x ) = − , namely a n , n , is given by (6.1) with all signs positive,we have(6.2) l , n = l n , n = − v g ( v g ( · · · v g n − ( v g n ( − ) · · · ) . The next lemma shows that l , n can be evaluated in terms of d n : = g g · · · g n . Lemma 6.2.
For each g with < g k ≤ / and for all n ∈ N we have d n ≤ l , n ≤ ( p / ) d n . Proof.
Clearly, 1 − v g ( t ) = + v g ( t ) g ( − t ) . Repeated application of this to (6.2) gives therepresentation l , n = κ n ( g ) d n , where κ n ( g ) is equal to21 + v g ( v g ( · · · v g n ( − ) · · · ) + v g ( · · · v g n ( − ) · · · ) · · · + v g n ( − ) . Since v / ( t ) ≤ v g ( t ) ≤ , we have 1 ≤ κ n ( g ) ≤ κ n ( / ) , where the last denotes the valueof κ n in the case when all g k = / . This gives the left part of the inequality. Let C n be thedistance between 1 and the rightmost extrema of T n . Hence, see e.g. p.7. of [28], C n = − cos ( p / n ) < p / ( · n ) . On the other hand, C n = κ n ( / ) − n . Therefore, κ n ( / ) < p / , and the lemma follows. (cid:3) For the case g n ≤ /
32 for all n , smoothness of the Green’s function for C \ K ( g ) and re-lated properties are examined in [18], [19]. The next theorem is complementary to Theorem1 of [18] and examines the smoothness of the Green function as g n → / Theorem 6.3.
The function G C \ K ( g ) is H¨older continuous with the exponent / if and onlyif (cid:229) ¥ k = e k < ¥ .Proof. Let us assume that (cid:229) ¥ k = e k < ¥ . Then (cid:213) ¥ k = ( − e k ) = a for some 0 < a < , d n = − n (cid:213) nk = ( − e k ) > a − n and, by Lemma 6.2, 2 a · − n ≤ l , n for all n ∈ N . Let z be an arbitrary point of K ( g ) . We claim that m K ( g ) ( D t ( z )) ≤ √ √ a √ t for all t > . It is evident for t ≥ / , as m K ( g ) is a probability measure. Let 0 < t < / . Fix n with l , n < t ≤ l , n − . We have t > a · − n . On the other hand, D t ( z ) can contain points from at most 4 basic intervals of level n − m F − n ([ − , ]) → m K ( g ) , by [30], we have m K ( g ) ( I j , k ) = / k for all k ∈ N and 1 ≤ j ≤ k .Therefore, m K ( g ) ( D t ( z )) ≤ − n < p t / a , which is our claim. The optimal smoothnessof G C \ K ( g ) follows from Theorem 6.1.Conversely, suppose that, on the contrary, (cid:229) ¥ k = e k = ¥ . This is equivalent to the condition4 n d n → n → ¥ . Thus, for any s >
0, there is a number N such that n > N impliesthat 4 n d n < s . For any t ≤ l , N + , there exists m ≥ N + l , m + < t ≤ l , m . Then, m K ( g ) ( D t ( )) ≥ m K ( g ) ( I , m + ) = − m − . On the other hand, by Lemma 6.2, t ≤ p s − m − .Therefore, for any t ≤ l , N + we have √ t p √ s ≤ m K ( g ) ( D t ( )) . Hence, the inequality √ p √ s √ r ≤ Z r m K ( g ) ( D t ( )) t dt , holds for r ≤ l , N + . By Theorem 6.1, G C \ K ( g ) ( − r ) ≥ √ p √ s √ r . Since s is here as small aswe wish, the Green function is not optimally smooth. (cid:3)
7. P
ARREAU -W IDOM SETS
Parreau-Widom sets are of special interest in the recent spectral theory of orthogonal poly-nomials. For different aspects of the theory, we refer the reader to the articles [12, 17, 32, 36]among others.A compact set K ⊂ R which is regular with respect to the Dirichlet problem is called a Parreau-Widom set if PW ( K ) : = (cid:229) j G C \ K ( c j ) < ¥ where { c j } is the set of critical pointsof G C \ K , which, clearly, is at most countable. A Parreau-Widom set has always positiveLebesgue measure, see [12].Our aim is to give a criterion when K ( g ) is a Parreau-Widom set. Note that, since au-tonomous Julia-Cantor sets in R have zero Lebesgue measure (see e.g. Section 1.19. in[21]), such sets can not be Parreau-Widom.We begin with a technical lemma. Lemma 7.1.
Given p ∈ N , let b = and b k + = b k ( + − p + k b k ) for ≤ k ≤ p − . Thenb p < . Proof.
We have b = + − p , b = + ( + ) − p + · · − p + · − p , · · · , so b k = (cid:229) N k n = a n , k − np with N k = k − a , k =
1. Let a n , k : = n > N k . The definition of b k + gives the recurrence relation(7.1) a n , k + = a n , k + k n (cid:229) j = a n − j , k a j − , k for 1 ≤ n ≤ N k + . If N k < n ≤ N k + , that is n = N k + m with 1 ≤ m ≤ N k + , then the formula takes the form a n , k + = k (cid:229) n − m + j = m a n − j , k a j − , k , since a n − j , k = j < m and a j − , k = j > n − m + . In particular, a N k + , k + = k a N k , k and a , k + = a , k + k . Therefore, a , k = + + · · · + k − < k / . Let us show that a n , k < C n nk with C n = − n / n ≥ . This gives the desired result,as b p = (cid:229) N p n = a n , p − np < + / · (cid:229) N p n = − n < . RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 15
By induction, suppose the inequality a j , k < C j jk is valid for 1 ≤ j ≤ n − k > . We consider j = n . The bound a n , i < C n ni is valid for i = , as a n , = n ≥ . Suppose it is valid as well for i ≤ k . We use (7.1) repeatedly, in order to reduce the second index, and, after this, the inductionhypothesis: a n , k + = k (cid:229) q = q n (cid:229) j = a n − j , q a j − , q < k (cid:229) q = nq n (cid:229) j = C n − j C j − < k (cid:229) q = nq < C n n ( k + ) , where C : = . Therefore the desired bound is valid for all positive n and k . (cid:3) Theorem 7.2. K ( g ) is a Parreau-Widom set if and only if (cid:229) ¥ k = √ e k < ¥ .Proof. Let E n = { z ∈ C : | F n ( z ) | ≤ } . Then G C \ E n ( z ) = − n log | F n ( z ) | . Clearly, the criticalpoints of G C \ E n coincide with the critical points of F n and thus they are real. Let Y n = { x : F ′ n ( x ) = } , Z n = { x : F n ( x ) = } . Clearly, Y n ∩ Z n = /0 and Z k ∩ Z n = /0 for n = k . Since F ′ n = F n − F ′ n − / g n , we have Y n = Y n − ∪ Z n − , so Y n = Z n − ∪ Z n − ∪ · · · ∪ Z , where Z = { } . We see that Y n ⊂ Y n + , so the set of critical points for G C \ K ( g ) is ∪ ¥ n = Z n and PW ( K ( g )) = (cid:229) ¥ n = (cid:229) z ∈ Z n − G C \ K ( g ) ( z ) . In addition, for each k ≥ n the function F k is constant on the set Z n − which contains 2 n − points. Let s n = n − G C \ K ( g ) ( z ) , where z is any point from Z n − .Then(7.2) PW ( K ( g )) = ¥ (cid:229) k = s k . We can assume that (cid:229) ¥ k = e k < ¥ . Indeed, it is immediate if (cid:229) ¥ k = √ e k < ¥ . On the otherhand, if z ∈ Z n − , that is F n − = , then F n ( z ) = − / g n = − − e n − e n . Since G C \ E n ր G C \ K ( g ) , we have s n > / | F n ( z ) | > / ( + e n ) > e n , as log ( + t ) > t / < t < . Therefore the supposition PW ( K ( g )) < ¥ implies, by (7.2), that (cid:229) ¥ k = e k < ¥ . Let a = (cid:213) ¥ k = ( − e k ) . By the remark above, 0 < a < . Our aim is to evaluate s n fromboth sides for large n . Let us fix N ∈ N such that n > N implies that e n ≤ a / . We consideronly such n after this point of the proof. Then 1 − e n > / s n : = e n − e n there exists p ∈ N such that(7.3) a · − − p < s n ≤ a · − p . Consider the function f ( t ) = b ( t − ) + t > , where b = / − e with e < / . Thus, F k + ( z ) = f ( F k ( z )) for b = g k + . If t = + s for small s , then we will use therepresentation f ( t ) = + s with 4 s < s = s + s / − e . Also, for each t ≥ t ≤ f ( t ) < b t < t . Let us fix z ∈ Z n − . Then, as above, | F n ( z ) | = + s n . Then F n + ( z ) = + s n + with4 s n < s n + = s n + s n / − e n + . We continue in this fashion to obtain F n + p ( z ) = + s n + p with(7.4) 4 p s n < s n + p = p s n · n + p − (cid:213) k = n + s k / − e k + . After that we use the second estimation for f . This gives F n + p ( z ) ≤ F n + p + ( z ) < F n + p ( z ) and, for each k ∈ N , F k n + p ( z ) ≤ F n + p + k ( z ) < ( / ) k − F k n + p ( z ) . From this, we have2 − n − p log F n + p ( z ) ≤ G C \ E n + p + k ( z ) ≤ − n − p [ log ( / ) + log F n + p ( z )] . Recall that G C \ E n + p + k ( z ) ր G C \ K ( g ) ( z ) , as k → ¥ and s n = n − G C \ K ( g ) ( z ) . Hence,2 − p − log F n + p ( z ) ≤ s n ≤ − p − [ log ( / ) + log F n + p ( z )] . Now suppose that K ( g ) is a Parreau-Widom set, so, by (7.2), the series (cid:229) ¥ k = s k converges.Then, by (7.4), we have s n ≥ − p − log ( + p s n ) . By (7.3), 4 p s n < ( + p s n ) > p s n / . Therefore, s n ≥ p s n / . We use (7.3) once again to obtain s n ≥ √ a s n / , whichimplies the convergence of (cid:229) ¥ k = √ e k . Conversely, suppose that (cid:229) ¥ k = √ e k < ¥ . Then s n ≤ − p log ( / ) + − p − s n + p . By (7.3),the first summand on the right is the general term of a convergent series. For the addend wehave 2 − p − s n + p < / a · p s n n + p − (cid:213) k = n ( + s k / ) , by (7.4). From (7.3) it follows that 2 p s n ≤ √ a s n < √ a e n , as e n < / . Let us show that(7.5) n + p − (cid:213) k = n ( + s k / ) < . This will give the estimation 2 − p − s n + p < p e n / a , where the right part is the general termof a convergent series. Then (cid:229) ¥ k = s k < ¥ , which is the desired conclusion, by (7.2).Thus, it remains to prove (7.5). We use notations of Lemma 7.1. By (7.3), we have1 + s n / ≤ + a − p / < b . Then,1 + s n + / < + a − e n + − p + ( + s n / ) < + − p + b = b / b and ( + s n / )( + s n + / ) < b . Similarly, by (7.4) and (7.3),1 + s n + k + / < + a ( − e n + ) · · · ( − e n + k ) − p + k b k < b k + / b k for k ≤ p − . Lemma 7.1 now yields (7.5). (cid:3) R EFERENCES [1] Alpan, G: Chebyshev polynomials on generalized Julia sets, Comput. Methods Funct. Theory, (2015),doi: 10.1007/s40315-015-0145-8[2] Alpan, G., Goncharov, A.: Orthogonal polynomials for the weakly equilibrium Cantor sets, Proc. Amer.Math. Soc., electronically published on May 6, 2016, DOI: http://dx.doi.org/10.1090/proc/13025 (toappear in print).
RTHOGONAL POLYNOMIALS ON GENERALIZED JULIA SETS 17 [3] Axler, S., Bourdon, P., Ramey, W.: Harmonic function theory, Second edition, Graduate Texts in Math-ematics, 137. Springer-Verlag, New York, (2001)[4] Barnsley, M.F.,Geronimo, J.S., Harrington, A.N.: Orthogonal polynomials associated with invariantmeasures on Julia sets, Bull. Amer. Math. Soc. (N.S.) (2), 381–384 (1982)[5] Barnsley M.F., Geronimo, J.S., Harrington, A.N.: Infinite-dimensional Jacobi matrices associated withJulia sets, Proc. Amer. Math. Soc. (4), 625–630 (1983)[6] Bessis, D: Orthogonal polynomials Pad´e approximations, and Julia sets, in: Orthogonal Polynomials:Theory & Practice, 294 (P. Nevai ed.), Kluwer, Dordrecht, 55–97 (1990)[7] Białas-Cie˙z, L.: Smoothness of Green’s functions and Markov-type inequalities, Banach Center Publ. , 27–36 (2011)[8] Brolin, H.: Invariant sets under iteration of rational functions, Ark. Mat. (2), 103–144 (1965)[9] Br¨uck, R.: Geometric properties of Julia sets of the composition of polynomials of the form z + c n , Pac.J. Math. , 347–372 (2001)[10] Br¨uck, R., B¨uger, M.: Generalized iteration, Comput. Methods Funct. Theory , 201–252 (2003)[11] B¨uger, M.: Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dyn. Syst. , 1289–1297 (1997)[12] Christiansen, J.S.: Szeg˝o’s theorem on Parreau-Widom sets, Adv. Math. , 1180–1204 (2012)[13] Comerford, M.: Hyperbolic non-autonomous Julia sets, Ergodic Theory Dyn. Syst. , 353–377 (2006)[14] Finkelshtein, A. M.: Equilibrium problems of potential theory in the complex plane. Orthogonal poly-nomials and special functions, Lecture Notes in Math., 1883, Springer, Berlin, 79117 (2006)[15] Fornæss, J.E., Sibony, N.: Random iterations of rational functions, Ergodic Theory Dyn. Syst. , 687–708 (1991)[16] Geronimo, J.S., Van Assche, W.: Orthogonal polynomials on several intervals via a polynomial mapping,Trans. Amer. Math. Soc. , 559–581 (1988)[17] Gesztesy, F., Zinchenko, M.: Local spectral properties of reflectionless Jacobi, CMV, and Schr¨odingeroperators, J. Differ. Equations. , 78–107 (2009)[18] Goncharov, A.: Best exponents in Markov’s inequalities, Math. Inequal. Appl. , 1515–1527 (2014)[19] Goncharov, A.: Weakly equilibrium Cantor type sets, Potential Anal. , 143–161 (2014)[20] Kosek, M.: H¨older exponents of the Green functions of planar polynomial Julia sets, Ann. Mat. PuraAppl. , 359–368 (2014)[21] Lyubich, M.: The dynamics of rational transforms: the topological picture, Russ. Math. Surv. (4),43–118 (1986)[22] Ma˜n´e, R., Da Rocha, L.F.: Julia sets are uniformly perfect, Proc. Amer. Math. Soc. (1), 251–257(1992)[23] Mayer, V., Skorulski, B., Urba´nski, M.: Regularity and irregularity of fiber dimensions of non-autonomous dynamical systems, Ann. Acad. Sci. Fenn. Math. , 489–514 (2013)[24] Mead, D. G.:Newton’s identities, Amer. Math. Monthly. , 749–751 (1992)[25] Milnor, J.: Dynamics in one complex variables, Princeton Universty Press, Annals of MathematicsStudies, , Princeton University Press, Princeton, NJ, (2006)[26] Peherstorfer, F., Volberg, A., Yuditskii, P.: Limit periodic Jacobi matrices with a prescribed p -adic hulland a singular continuous spectrum. Math. Res. Lett. , 215–230 (2006)[27] Ransford, T.: Potential theory in the complex plane, Cambridge University Press, (1995)[28] Rivlin, T.J.: Chebyshev polynomials : from approximation theory to algebra and number theory, SecondEdition, J. Wiley and Sons, New York, (1990)[29] Saff, E.B., Totik, V.: Logarithmic potentials with external fields, Springer-Verlag, New York (1997)[30] Simon, B.: Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging , 713–772(2007)[31] Simon, B.: Szeg˝o’s Theorem and Its Descendants: Spectral Theory for L Perturbations of OrthogonalPolynomials, Princeton University Press, Princeton, NY (2011)[32] Sodin, M., Yuditskii, P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. , 387–435 (1997) [33] To´okos, F., Totik, V.: Markov inequality and Green functions, Rend. Circ. Mat. Palermo math. 2 Suppl. , 91–102 (2005)[34] Totik, V.: Metric properties of harmonic measures, Mem. Am. Math. Soc. (2006)[35] Van Assche, W.: Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, 1265,Springer-Verlag, Berlin (1987)[36] Volberg, A., Yuditskii, P.: Kotani-Last problem and Hardy spaces on surfaces of Widom type, Invent.Math. , 683–740 (2014)D EPARTMENT OF M ATHEMATICS , B
ILKENT U NIVERSITY , 06800 A
NKARA , T
URKEY
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , B
ILKENT U NIVERSITY , 06800 A
NKARA , T
URKEY
E-mail address ::