WEIGHTED KNEADING THEORY OF ONE-DIMENSIONAL MAPS WITH A HOLE

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.


Introduction.
Let I ⊂ R be a compact interval and F a linear expanding map on I.We study the dynamical system given by iterating points by F .Of particular interest is the set defined by the points that remain in the domain of F under iteration.An equivalent way of viewing this situation is as the inverse of an iterated function system (IFS) (see [4]).Let f = {f i } n i=1 be an IFS, a collection of self-maps on I, defined by f i (x) := ρ i x + i , i= 1,...,n, ( where for all i, 0 < |ρ i | < 1 and i ∈ R. Let E be the corresponding self-similar set, the attractor.If f i is monotone, then it is usual to see E as the repeller of a linear expanding map F : n i=1 f i (I) → I, which will be denoted by F = (F 1 ,...,F n ), where We consider the piecewise linear map F with a single hole, that is, there is an open subinterval I h ⊂ I with I h ≠ ∅ such that I is the disjoint union of I h and n i=1 Im(f i ) (see [8,10]).The points x ∈ I h will be mapped out of I and the same will happen to all the points x ∈ F −k (I h ) for k ≥ 1.The set k F −k (I h ) is open and dense in I and has full Lebesgue measure (see [14]).
The hole and the set of n laps of F determines a partition ᏼ I := {I 1 ,...,I h ,...,I n } of the interval I. Considering the orbits of the lateral limit points of the discontinuity points and turning points, we define a Markov partition ᏼ I of I.
The outline of the paper is as follows.In Section 2, we develop a weighted kneading theory to expanding discontinuous maps with a hole.In this section, we give a brief presentation of the kneading theory associated to F .For more details, see [6], and for maps of the interval with holes, see [9].The new kneading approach using weights is inspired on [8], where we compute explicitly the escape rate that is characterized by a conditional invariant measure.The weights introduced in the kneading theory are defined by the inverses of the derivatives of the iterates of the discontinuity points and turning points of F , associated to a real parameter β.We consider the transfer or Perron-Frobenius operator L φ associated to the map F and to the Markov partition ᏼ I .The transfer operator has a matrix representation, which we will denote by Q β .This matrix can be viewed as the matrix of L φ acting on a finite-dimensional vector space of functions.
It is known that the spectrum of the transfer operator determines the ergodicity of the dynamical system.In Section 3, we will give an algorithm to compute this spectrum and to relate the transfer operator with the weighted kneading determinant D(t, β) (see Theorem 3.6).To establish this relation, we introduce a weighted matrix V β and we use complexes and homology with weights (see Theorem 3.5).These results allow us to prove the main result.
Theorem 1.1.Suppose that the kneading data associated to an expanding discontinuous map with a hole F correspond to periodic, eventually periodic orbits or to orbits that lie in the hole.Let D(t, β) be the weighted kneading determinant. (i We remark that we can obtain the same results for a finite union of disjoint holes I h j ⊂ I.
To simplify the presentation, we consider the points x (1) and x (2n) as fixed points.To the orbit of each point x (j) , with j = 2,...,2h−2, 2(h+1)−1,...,2n−1, we associate a sequence of symbols S (j) given by S (j) where (2.5) We denote by Ꮽ the ordered set of n + 1 symbols, corresponding to the laps and the hole of F , that is, and according to the real-line order, We designate by Ꮽ N the space of all sequences of symbols on the alphabet Ꮽ.
Definition 2.1.The kneading data for the map F is the (2n − 4)-tuple of symbolic sequences (2.8) The kneading increments introduced in [6] are defined by formal power series with coefficients in Z[[t]], the subring of the ring Q[[t]].For maps of the interval with a hole and more than one discontinuity point and turning points, we have several kneading increments, whose number depends on the number of discontinuity points and turning points of the map F (see [9]).The kneading increments are defined by (2.9) In the case where a i is an endpoint of the hole I h , the increments are defined by where θ a i (t) is the invariant coordinate of each symbolic sequence associated to the itinerary of each point a i , with 1 < i < n + 1.Each lateral invariant coordinate is defined by where and S (j) k is the sequence of symbols corresponding to the orbits of a ± i .The increments ν a i (t), with 1 < i < n+ 1, can also be written in the following way: where the coefficients This matrix is called the kneading matrix associated to the map F .The kneading determinant is denoted by D(t).Now we are going to present the main definition of this paper, that is, the characterization of the weighted invariant coordinates.This definition allows us to construct a weighted kneading theory similar to the previous one.Definition 2.2.For the kneading data of the map F , the weighted invariant coordinate of each point a i , with 1 < i < n+ 1 and β ∈ R, is defined by where τ 0 (a ± i ) := 1 and for k > 0, Note that the derivative of the map F satisfies the condition inf |F i (x)| > 1 on each interval f i (I).For each point a i , with i = 1,...,h−1,h+2,... ,n+1, the weighted kneading increment is defined by (2.16) For the endpoints of the hole I h , the weighted increments are defined by (2.17) Separating the terms associated to the symbols on the alphabet Ꮽ, the weighted increments ν a i (t, β) are written in the following way: which we will call the weighted kneading matrix associated to F .The determinant of this matrix will be called the weighted kneading determinant and will be denoted by D(t, β).
Remark 2.3.To an eventually periodic orbit of a point x (j) represented by corresponds the weighted cyclotomic polynomial where q is the period of the orbit.If the orbit is periodic, then the weighted cyclotomic polynomial is be the set of the points correspondent to the orbits of the lateral limit points of the discontinuity points and turning points ordered on the interval I.This set allows us to define a subpartition ᏼ I of ᏼ I = {I 1 ,...,I h ,...,I n }.The subpartition with m ≥ n, determines a Markov partition of the interval I.Note that the hole is an element of the Markov partition.Note also that F determines ᏼ I uniquely, but the converse is not true.
The IFS f induces a subshift of finite type whose m × m transition matrix A = [a ij ] is defined by We remark that if there exist k points b i such that b i ∈ int I h , with 1 < i < m + 1, then the matrix A has k + 1 columns with all elements equal to zero, correspondent to the hole.
We denote this subshift by (Σ A ,σ ), where σ is the shift map on ..,m} correspondent to the m states of the subshift.Concerning this subshift (Σ A ,σ ) and the associated Markov partition ᏼ I , we consider a Lipschitz function φ : I → R defined by where This function is a weight for the dynamical system associated to the subshift, depending on the real parameter β (compare with [11]).Let ᏸ 1 (I) be the set of all Lebesgue integrable functions on I.The transfer operator L φ : ᏸ 1 (I) → ᏸ 1 (I), associated with F and ᏼ I , is defined by where χ I j is the characteristic function of I j .Note that by definition of F , F −1 j (x) = f j (x), with x ∈ F(I j ).Note also that, for any Borel subset J ⊂ I, we have where the sets {f j (F (I j ) ∩ J)} n j=1 are mutually disjoint.Depending on J, the set f j (F (I j ) ∩ J) can be empty.Now, we will restrict our attention to the transfer operator associated with F and to the Markov partition ᏼ I .Given J i ∈ ᏼ I , let Y 1i ,...,Y ki be the preimages of J i under F , that is, (2.30) Then, we can define continuous maps f j | J i := Ψ ji : J i → Y ji that correspond to the IFS f restricted to the interval J i such that y j = Ψ ji (x) are the preimages of x ∈ J i .Thus, for each x ∈ J i , we have where Nevertheless, for each interval J j ∈ ᏼ I , we consider where a ij are the entries of the transition matrix A, with 1 ≤ i, j ≤ m.By formula (2.31), we can write (2.34) In this paper, we consider a class of one-dimensional transformations that are piecewise linear Markov transformations.Consequently, the transfer operator has the following matrix representation.Let Ꮿ be the class of all functions that are piecewise constant on the partition ᏼ I .Thus, for some constants π 1 ,...,π m .We remark that g will also be represented by the column vector π g = (π 1 ,...,π m ) T .Using formula (2.34) and considering g ∈ Ꮿ with g = χ J k for some 1 ≤ k ≤ m, the transfer operator L φ has the following matrix characterization: for the weighted dynamical system associated to (Σ A ,σ ).If D β is the diagonal matrix defined by and A is the transition matrix, then the matrix Q β is the m × m weighted transition matrix defined by The entries of this matrix are where the derivative F j is evaluated on the interval J j of the partition ᏼ I .We refer to [3,13] and the references therein to other important spectral properties of the transfer operator, and [4] for this operator with respect to the cookie-cutter system.In [8], we use the matrix Q β with β = 1 to compute the escape rate and the conditional invariant measure which generates the unique invariant probability measure.
There is an isomorphism between (Σ A ,σ ) and (ᏼ I ,F) (see [15]).If w = (i 0 i 1 •••) and w = (i 0 i 1 •••) are two points of Σ A , then we consider the Markov partition defined by ᏼ I := Σ A / ∼, where w ∼ w if and only if i 0 = i 0 .Using this isomorphism, we consider the trace of the transfer operator defined by where Fix(F ) denotes the set of fixed points of F .We consider the pressure function of φ(x) = log |F (x)| −β as β varies, P (β), defined by where Fix(F k ) denotes the set of fixed points of F k (see [4,12,15]).Thus, exp P (β) is the largest eigenvalue λ β of the transfer operator L φ , which is equal to the spectral radius of the matrix Q β (see [13]).Nevertheless, it is known that the pressure can be characterized by the variational principle as the supremum over all invariant probability measures on E. In this case, the supremum is attained by the weighted Markov measure µ β , that is, the measure µ β is the unique measure that maximizes this expression.See [8] for the definition of µ β , the weighted metric entropy h µ β (F ), and the weighted Lyapunov exponent χ µ β (F ) with respect to this measure.We remark that the weighted zeta function for a weighted subshift of finite type is given by

.43)
For more discussions about the zeta function for a subshift of finite type without weights, see [2], and for another approach with weights, see [1,7].

Lemma 3.2. The next diagram is commutative
Proof.Let the intervals J j , with 1 ≤ j ≤ m, be represented by the column vector J j = (0,...,0,χ J j , 0,...,0) T , where χ J j is in the jth-position, that is, as a function in Ꮿ.Then, one has where u lj are the nonzero elements of the jth column of the matrix BA.The above equalities make a description of the transition of the interval J j by the border of the intervals J k such that F(int J j ) ⊇ int J k , weighted by |F (J j )| −β .On the other hand, we have Consider that y (i) is a point associated to a turning point or to a discontinuity point of F and F(y (i) ) = y (s) , with s < i.Consequently, (3.9) If there exist z 1 turning points or discontinuity points between y (s) and y (i−1) , then we have pairs of consecutive points y (k l −1) , y (k l ) , with s < k l − 1, k l < i− 1 and 1 ≤ l ≤ z 1 such that Suppose that y (i+1) ∈ int I p , with 1 ≤ p ≤ n, that is, y (i+1) ∈ Ᏻ and F(y (i+1) ) = y (r ) , with r > i+ 1.In this case, we have Similarly, if there exist z 2 turning points or discontinuity points between y (i+1) and y (r ) , then we have pairs of points y (kw ) , y (kw +1) , with i + 1 < k w , k w + 1 < r and 1 ≤ w ≤ z 2 such that As the weight is constant on each interval J j , we get where the pairs of points y (l) , y (l+1) lie in the set y (s) ,y (s+1) ,...,y (k l ) ,y (k l +1) ,...,y (i−1) ,y (i) ,...,y (kw ) ,y (kw +1) ,...,y (r −1) ,y (r ) (3.14) and describe the border of the intervals J k such that F(int J j ) ⊇ int J k .From this, it follows that The proof of the remaining cases is similar according to the behavior of F and the above definition.
Let H 0 := C 0 /B 0 , where B 0 = ∂(C 1 ) is a subspace of C 0 .Note that two consecutive laps without a discontinuity point between them are considered as two connected components.The map ζ : C 0 → H 0 associates to each point y (i) , with 1 ≤ i ≤ q, the respective interval I j , with 1 ≤ j ≤ n.This map is represented by the q ×n matrix U = [u ij ], where u ij := 1 if the point y (i) lies in I j and all the remaining entries of the matrix are zero.Lemma 3.3.If y (i 1 ) and y (i 2 ) are two points on the interval I j with 1 ≤ j ≤ n, then UV β y (i 1 ) T = UV β y (i 2 ) T .
The above lemma suggests the next definition and result.Associated to each matrix V β , we have only one map ξ : H 0 → H 0 which reflects the monotonicity of F .The map ξ is represented by the where The main results can now be stated.The next theorems establish the relation between the weighted transition matrix, the weighted matrix V β , and the weighted kneading determinant.
Theorem 3.5.Under the conditions of the previous lemmas, the following relation holds between the characteristic polynomials of the matrices Q β , V β , and K β : (3.25) Proof.The statement is a consequence of the above lemmas and according to some homological algebra results (see [5]).
Theorem 3.6.If the kneading data associated to an expanding discontinuous map with a hole F corresponds to periodic, eventually periodic orbits, or to orbits that lie in the hole, then the weighted kneading determinant is given by where R(t) is a product of weighted cyclotomic polynomials correspondent to those periodic or eventually periodic orbits.
It is obvious that this statement strongly depends on the number of laps and the kneading data associated to F .For this reason, the analysis of the general situation is difficult.We will prove the statement for a map F = (F 1 ,F 2 ).The general case follows in a similar way.
and Ꮽ = {L, H, R}.The orbits of the points a + 1 and a − 4 can be periodic, eventually periodic, or lie in the hole.We consider that the kneading data associated to this map is given by o x (2) ,o x (5) = LS (2) where p and q are the periods of the orbits.The weighted kneading increments are where (5) t i S (5) i 1 − τ q x (5) t q . (3.30) If we write and analogously for L q and R q , then we have The weighted kneading determinant D(t, β) for these kneading data is Let det Θ β = det(I − tΘ β ) be the characteristic polynomial of the matrix Θ β , where I is the identity matrix.Thus, we have where and similarly for δ 2,k 1 and δ 5,k 2 .Using matrix elementary operations for the matrix I − tΘ β , we have the following equivalent matrix: Now we will compare the elements of the above matrix to the elements of the weighted kneading matrix N(t, β).Note that (3.37) On the other hand, In particular, if µ 2,p−1 ≠ 0, then x (2) p−1 is associated to the symbol R, that is, Consequently, in (3.37), we have Hence, in (3.38), the fact that the orbit is periodic implies that we return to the symbol L. Thus, we have Let R(t) be the product of cyclotomic polynomials and P K β (t) the characteristic polynomial of the matrix K β associated to F .Set (3.43) Using the above comparison between the elements of the equivalent matrix to I − tΘ β and the elements of the weighted kneading matrix, we have The main result, Theorem 1.1, will allows us to compute explicitly the Hausdorff dimension, the escape rate, and the topological entropy.
Proof of Theorem 1.1.Considering the transfer operator given in (2.34), we have Let a ij be the entries of the transition matrix A. For each J i ∈ ᏼ I , with 1 ≤ i ≤ m and β ∈ R, the eigenvalue equation corresponding to an eigenvalue λ β is for the operator L φ characterized by the matrix Q β .According to [11] and using (2.41), the largest eigenvalue of the transfer operator is exp P (β).Hence, expP (β) is the spectral radius λ β of the matrix Q β .
On the other hand, considering the parameter β = 1, we have that λ 1 = exp P (1) is the largest eigenvalue of the matrix Q 1 .The second statement follows from [14], where the escape rate γ is given by γ = −P (1).Thus, the escape rate is γ = log(λ −1 1 ), where λ −1 1 = t 1 is the least real positive solution of P Q 1 (t) = 0.If β = 0, then the determinant D(t, 0) corresponds to the kneading determinant described in [9], where t −1 0 = λ 0 is the growth number of F , that is, the spectral radius of the transition matrix A. Consequently, log(λ 0 ) is the topological entropy of the map F .Remark 3.7.The theory presented in this paper with respect to periodic, eventually periodic orbits, or to the orbits that lie in the hole is also valid for aperiodic orbits.In this case, the invariant coordinates associated to the turning points and to the discontinuity points are formal power series.The computation of the topological invariants is done by approximation using periodic, eventually periodic orbits, or the orbits that lie in the hole.
The above results are illustrated in the next example, showing in detail the techniques under discussion.