Wenxiu Ma is currently a graduate student of University of Science and Technology of China. Her research mainly focuses on one dimensional dynamics and complex dynamics
A special class of cubic polynomials possessing decay of geometry property is studied. This class of cubic bimodal maps has generalized Fibonacci combinatorics. For maps with bounded combinatorics, we show that they have an absolutely continuous invariant measure.
Graphical abstract
Abstract
A special class of cubic polynomials possessing decay of geometry property is studied. This class of cubic bimodal maps has generalized Fibonacci combinatorics. For maps with bounded combinatorics, we show that they have an absolutely continuous invariant measure.
Public Summary
We study the combinatorial properties of (r, t)-Fibonacci bimodal maps.
We construct an induced map G and show that G admits an acip.
The dynamical properties of unimodal interval maps have been extensively studied. The 'decay of geometry' property plays an essential role in the study of quadratic dynamics. Several results, including density of hyperbolicity and Milnor’s attractor problem, rely on this phenomenon.
For the unimodal case, let I0⊃I1⊃I2⊃… be the principal nest of f. The scaling factor of f is defined as μn:=|In+1||In|. Decay of geometry means that μni decreases to 0 exponentially fast for a subsequence ni. This concept first appeared in the work of Jakobson and Światiek[3] for non-renormalizable maps with negative Schwarzian derivatives. Lyubich[9], Graczyk and Światiek[2] solved it independently using complex techniques. These proofs make elaborate use of complex methods and do not seem to work for critical orders smaller than 2. More recently, Shen[12] used real analysis techniques to prove the decay of geometry property for smooth unimodal maps with a critical order of no more than 2, thus solving the Milnor attractor problem in this case.
However, the decay of geometry loses its universality for unimodal maps with a larger critical order, or even for multimodal maps with quadratic critical points. For the unimodal case, the typical example is the Fibonacci unimodal maps. It is well-known that Fibonacci unimodal maps possess decay of geometry and admit an absolutely continuous (with respect to the Lebesgue measure) invariant probability measure for critical order ℓ⩽2, and have bounded geometry for critical order ℓ>2, see Refs. [8, 6]. For multimodal maps, the real cubic polynomial with two nondegenerate critical points does not have uniform decay of geometry property either. A preliminary investigation of this phenomenon was carried out by Światiek and Vargas[14], who constructed two cubic polynomials, one with bounded geometry and the other with decay of geometry.
To date, multimodal maps are rarely known. The principal nest is a useful tool when studying the geometric properties of interval maps, but it seems inonvenient for treating metric problems in multimodal cases. Unlike the unimodal case, the scaling factors fail to give distortion control of the first return map of the principal nest. However, in Ref. [15], Vargas constructed the Fibonacci bimodal map using a new tool, named 'twin principal nest', which we will explain later. Our recent work[5] showed that a wide class of cubic maps have the 'decay of geometry' property in the sense that the ratio of twin principal nests decreases at least exponentially fast. In this paper, we concentrate on the metric properties of generalized Fibonacci bimodal maps with uniformly bounded combinatorics.
1.1
Preliminaries
For convenience, (a,b) denotes the interval with endpoints a and b, even though a>b. For example, let (2,1) refer to (1,2). If J and J′ are two intervals on the real line, by J<J′(J⩽J′), we mean that y<y′(y⩽y′) for every y∈J and y′∈J′; analogously, we define a<J and a⩽J′ for real number a.
Denote I=[0,1]. A continuous map f:I→I is called bimodal if:
1.f({0,1})={0,1};
2. there exist exactly two points c<d (called turning points) that are the local extreme of f;
3.f is strictly monotone on subintervals determined by these points.
If the points {0,1} are fixed, then we say that the bimodal map f is positive, and in the case that these points are permuted, we say that f is negative. Examples of bimodal maps are parameterized families of real cubic polynomials P+ab and P−ab given by P+ab(x)=ax3+bx2+(1−a−b)x and P−ab(x)=1−ax3−bx2−(1−a−b)x. We are mainly interested in bimodal maps that have neither periodic attractors nor wandering intervals.
For T⊂I, let D(T)={x∈I:fk(x)∈Tforsomek⩾1}. The first entry mapRT:D(T)→T is defined as x→fk(x)(x), where k(x) is the entry time of x into T, i.e., the minimal positive integer such that fk(x)(x)∈T. The map RT|(D(T)∩T) is called the first return map of T. A component of D(T) is called an entry domain of T and a component of D(T)∩T is called a return domain.
An open set J⊂I is called nice if fn(∂J)∩J=∅ for all n⩾0. Let T⊂I be a nice interval. Let Lx(T) denote the entry domain of T containing x.
A point x∈I is called recurrent provided x∈ω(x).
Let B denote the collection of C3 bimodal maps f:I→I, which have no wandering intervals and all periodic cycles of hyperbolic repelling. Let Crit(f) denote the set of critical points of f, i.e., the set of points where Df vanishes. Note that {c,d}⊂Crif(f). Let B+ and B− denote the subset of positive and negative bimodal maps, respectively, from class B. If f∈B+, then there exists a fixed point p between c and d; otherwise, ∂I contains an attracting fixed point. Let p1<p2 be such that f(p1)=f(p2)=p. Define I0=(p1,p), J0=(p,p2). If f∈B−, we discuss three cases:
1.f has three fixed point in (0,1). In this case, there exists a fixed point p in (c,d), then define I0 and J0 as above.
2.f has one fixed point p in (0,1) with three preimages {p,p1,p2} specified by p1<p2. If p<p1<p2, define I0=(p,p1)∋c and J0=(p1,p2)∋d; if p1<p2<p, define I0=(p1,p2) and J0=(p2,p).
3.f has one fixed point p in (0,1) with only one preimage, that is, f−1(p)={p}. This case can be reduced to the positive case since f2 restricted on [p,1] is always a positive bimodal map.
Assume that both c and d are recurrent. For every n⩾1, define In:=Lc(In−1∪Jn−1) and Jn:=Ld(In−1∪Jn−1) inductively. The two sequences of nested intervals
I0⊃I1⊃I2⊃…⊃{c}andJ0⊃J1⊃J2⊃…⊃{d}
are called the twin principal nest of f. The scaling factor of f is defined as
λn:=max{|In||In−1|,|Jn||Jn−1|}.
Let gn denote the first return map to In−1∪Jn−1. The restriction of gn on In and Jn are unimodal, while its restriction on any other branches are monotone and onto In−1 or Jn−1. The first return map gn is called a central return if gn(c)∈In∪Jn or gn(d)∈In∪Jn; otherwise, gn is called non-central return. In the case when gn is non-central, let In1 and Jn1 (possibly coincide) denote the first return domains intersecting {gn(c),gn(d)}.
Given f∈B, f is called combinatorially symmetric if there exists an orientation-reversing homeomorphism h:I→I such that h∘f=f∘h.
Note that any combinatorially symmetric maps can be quasisymmetrically conjugated to an odd function. Finally, let B∗ denote the collection of combinatorially symmetric bimodal maps from B with recurrent turning points satisfying ω(c)=ω(d). Class B∗ is nonempty since it contains infinitely renormalizable maps and Fibonacci bimodal maps.
1.2
Statement of results
Definition 1.1. A bimodal map f∈B∗ is called (r,t)-Fibonacci if:
(1)f(c),f(d)∉I0∪J0 while fi(c),fi(d)∈I0∪J0 for i=2,3;
(2)In1 and Jn1 are defined and disjoint for all n⩾1;
(3) for each n⩾1, In∪Jn⊂gn(In∪Jn);
(4) for each n⩾1, (ω(c)∪ω(d))∩(In−1∪Jn−1)⊂In∪In1∪Jn∪Jn1;
(5) for each n⩾1, gn|(In∪Jn)=grn−1|(In∪Jn) for some integer r⩾2;
(6) for each n⩾1, gn|(In1∪Jn1)=gtn−1|(In1∪Jn1) for some integer t⩾1.
A pair of integers (r,t) is called admissible if there exists (r,t)-Fibonacci bimodal maps. Actually, not all (r,t) are admissible; for example, there does not exist (4,2)-Fibonacci bimodal maps. It was proven in Ref. [5] that under some conditions, (r,t) is admissible (see Section 2 for detail). The admissible pair (r,t) is just a simplification of Admissibility condition A for stationary combinatorics.
Lemma 1.1. Any pair of integers (r,t) is admissible if either r is even, t is odd with t<r, or r is odd, t is odd with t<r.
Suppose f is (r,t)-Fibonacci, then we can say f has 'uniformly bounded combinatorics'.
According to Ref. [11], the families of real cubic polynomials P+ab and P−ab are 'full families'. Combined with the rigidity theorem developed in [7], we can obtain the following corollary.
Corollary 1.1. For any admissible pair (r,t), there exists exactly one (r,t)-Fibonacci bimodal map in P+ab, and one (r,t)− Fibonacci bimodal map in P−ab.
Let B denote the class of (r,t)-Fibonacci bimodal maps in P+ab∪P−ab.
Theorem 1.1.[5] Suppose f∈B, then there exist constants C=C(f)>0 and 0<λ=λ(f)<1 such that the scaling factor of f decreases at least exponentially: λn(f)⩽Cλn for all n⩾1.
The main result of this paper is the following theorem.
Theorem 1.2. For any f∈B, f admits an acip μ.
This paper is organized as follows. In Section 2, we study the combinatorial properties of (r,t)-Fibonacci bimodal maps. In Section 3, we construct an induced map G and study the metric properties of G. We show that G admits an acip and prove that for any f∈B, f has an acip, which is the main result of this paper. In Section 4, we give a conclusion of this paper.
2.
Combinatorics
In this section, we study the combinatorial properties of (r,t)-Fibonacci bimodal maps. For any (r,t)-Fibonacci bimodal map f, we can consider the map g1|(I0∪J0) and rescale the interval back to [0,1], which is the notion of renormalization. Repeating this procedure is called generalized renormalization. Comparing Ref. [2, Section 4], the analytic extension of a generalized renormalization gn can be treated as a type Ⅱ special box mapping. Because the maps we consider are combinatorially symmetric, the positions of In1,Jn1,In and Jn have some constraints. Hence, we define 3 types of g1, the types of gn are similar.
● Type A: if g1(I11)=J0, g1(I1c)⊂J0 and g1(J11)=I0, g1(J1d)⊂I0;
● Type B: if g1(I11)=J0, g1(I1c)⊂I0 and g1(J11)=I0, g1(J1d)⊂J0;
● Type C: if g1(I11)=I0, g1(I1c)⊂J0 and g1(J11)=J0, g1(J1d)⊂I0.
Let us subdivide each type A, B and C in subtypes AijBij and Cij with i,j∈{+,−}, where i=+ or i=− if the non-central branches of f are orientation-preserving or orientation-reversing, respectively, and j=+ or j=− if f is locally maximal or minimal at c, respectively. Finally, let A+=A++∪A+− and define A−,B+,B−,C+,C−,D+,D− analogously.
The proof of the following lemma can be found in Ref. [5].
Lemma 2.1. Suppose (r,t) is admissible, then there exists an (r,t)-Fibonacci bimodal map f, and its first return map sequence {gn} satisfies the following conditions:
(1) If r is even and t is odd, the first return map sequence {gn} exhibits the sequence A+B−C−A−B+C+A+⋯ or C−A−B+C+A+B−C−⋯.
(2) If r is odd and t is odd, the first return map sequence {gn} exhibits the sequence A+A+A+A+⋯ or C−A+A+A+⋯.
Let S1=2,ˆS1=1, for n⩾1, define inductively
Sn+1=Sn+(r−1)ˆSnandˆSn+1=Sn+(t−1)ˆSn.
Then the return times of critical points c and d to In−1∪Jn−1 are equal to Sn, while the return times of gn(c) and gn(d) to In−1∪Jn−1 are equal to ˆSn.
Example 2.1. The Fibonacci bimodal maps studied in Ref. [15] are (2,1)-Fibonacci, where the first return time of critical points c and d to In−1∪Jn−1 coincides with the Fibonacci sequence. In this case ˆSn=Sn−1 and hence Sn+1=Sn+Sn−1. In particular, the first return map sequence {gn} exhibits the sequence
A++B−+C–−A−+B+−C+−A+−B–C−+A–−B++C++A++⋯
or
C−+A–−B++C++A++B−+C–−A−+B+−C+−A+−B–−C−+⋯
depending on f∈B+ or f∈B−.
Example 2.2. For r=4 and t=3, we can find the (4,3)-Fibonacci bimodal maps, and the first return map sequence {gn} exhibits the sequence A+B−C−A−B+C+A+⋯ or C−A−B+C+A+B−C−⋯.
For r=5 and t=3, we can find the (5,3)-Fibonacci bimodal maps, and the first return map sequence {gn} exhibits the sequence A+A+A+A+⋯ or C−A+A+A+⋯.
3.
Acip for f
Following the strategy in Ref. [1], the idea is to construct a Markov induced map G over f with the intervals In and Jn as a countable set of ranges: G is defined on a countable collection of intervals Ti, G|Ti=fsi|Ti is a diffeomorphism and G(Ti)=In or G(Ti)=Jn for some n. We will construct a G-invariant measure ν≪Leb, and estimate ν(In) and ν(Jn), where ν≪Leb means that ν is an absolutely continuous (with respect to the Lebesgue measure) invariant probability measure (acip for short). One result of this section is the following proposition, we will give a proof in Subsection 3.4.
Proposition 3.1. The induced map G admits an acip ν. Moreover, for arbitrarily small ϵ>0, there exists C0=C0(f,ϵ) such that ν(In)+ν(Jn)⩽C0(√ϵ)n.
It is well-known that the nonexistence of a ‘wild attractor’ is based only on the decay of geometry in the unimodal case.
Corollary 3.1. Suppose f∈B, then f has no Cantor attractor.
Proof. This follows from the observation that a Cantor attractor has zero Lebesgue measure, and, disregarding the critical points, is invariant by G. Hence G cannot carry an acip if a Cantor attractor is present.
3.1
Distortion
Given a bounded interval I and a constant τ>0, let τI denote the open interval that is concentric with I and has length τ|I|. We say a bounded interval J is τ-well inside an interval T if (1+2τ)J⊂T, i.e. both components of T∖J have a length of at least τ|J|.
The distortion of a C1 function h:J→h(J) is defined as
D(h):=D(h;J):=supx,y∈Jlog|Dh(y)||Dh(x)|.
Let us say a diffeomorphism h:J→h(J) belongs to the distortion class FCp if it can be written as
Q∘φq∘Q∘φq−1∘⋯∘Q∘φ1
with q⩽p, where Q(x)=x2 and D(φj)⩽C for all 1⩽j⩽q.
The Schwarzian derivative of a C3 function ϕ:T→R is defined as
Sϕ:=ϕ‴ϕ′−32(ϕ″ϕ′)2.
It is well known that if Sϕ⩽0, then Sϕn⩽0 for all n⩾1. Moreover, if f is a real polynomial with only real critical points, then Sf<0; see, for example, Ref. [11, Chapter IV, Exercise 1.7].
We shall use the following version of the Koebe principle, which was proved in Ref. [11].
Proposition 3.2. Assume that h:T→h(T) is a C3 diffeomorphism with Sh<0. If J is a subinterval of T such that h(J) is κ-well inside h(T), then,
D(h;J)⩽logKκ,whereKκ=(1+κκ)2.
The interval J is κ′-well inside T, where κ′=κ2/(1+2κ).
Suppose f∈B. For any ϵ>0 small, pick a large positive integer n0=n0(f) such that λn⩽ϵ for all n⩾n0, and such that f|In0 and f|Jn0 can be written as x→φ∘x2 with D(φ)⩽1/4. By Proposition 3.2, it follows that for each n⩾n0, if J is a return domain to In or Jn, and fs|J is the return, then fs|J can be written as x→φ∘x2 with D(φ)⩽1/2 provided ϵ is sufficiently small.
3.2
Construction of induced maps
Let G0 be the first return map to I0∪J0. Then G0 has finite number of branches, the two central branches (each contains one critical point) are the branches with return time 2, and each non-central branch maps diffeomorphically onto I0 or J0 with return time 1.
We will construct a sequence of maps Gn:⋃iLn+1i∪⋃iRn+1i→I0∪J0 inductively such that
① ⋃iLn+1i⊂I0 and ⋃iRn+1i⊂J0 are finite unions and for n⩾1, Gn=Gn−1 outside In∪Jn;
② the central branches Ln+10=In+1 and Rn+10=Jn+1, Gn|In+1 and Gn|Jn+1 are the first return maps to In or Jn;
③ for each i≠0, there exists bi⩽n such that Gn:Ln+1i→Ibi or Gn:Ln+1i→Jbi is a diffeomorphism; analogously for Rn+1i;
④ for each i≠0, Ln+1i⊂In and ∂Ln+1i∩∂In≠0 imply Gn(Ln+1i)=I0orJ0 (and the common boundary of Ln+1i and In maps to the fixed point p); analogously for Rn+1i;
⑤ Gn(x)=fs(x) implies that f(x),…,fs−1(x)∉In∪Jn.
By definition G0 satisfies the above statements, so let us assume that by induction Gn exists with the above properties, and construct Gn+1. It suffices to construct Gn+1 inside In+1 since the construction inside Jn+1 is similar.
Set Gn+1(x)=Gn(x) for x∉In+1∪Jn+1. Let kn∈N be minimal so that Gknn(c)∈In+1∪Jn+1. Since all the returns are non-central, kn⩾2. Define K0=In+1, Kkn=In+2 and, for 0⩽j⩽kn−1, let Kj be the component of dom(Gj+1n) which contains c. Next define on Kj∖Kj+1,
Gn+1|In+2=Gknn|In+2 is the first return map to In+1∪Jn+1.
Properties ① and ② hold by construction for Gn+1. Property ③ holds because if Gj+1n(x)∈In+1 (resp. Jn+1) for some x∈In+1∖In+2 then Gn+1(Ln+1i)=In+1 (resp, Jn+1) for the corresponding domain Ln+1i∋x; and if Gj+1n(x)∉In+1∪Jn+1 then by the induction assumption Gn+1(Ln+1i) is equal to some Ib or Jb with b⩽n, because then Gn+1(x)=Gj+2n(x). Property ④ holds immediately because ∂In and ∂Jn are mapped by Gn into ∂I0∪∂J0. To show that property ⑤ holds, take x∈Kj∖Kj+1 and let y=Gjn(x). Note that Gjn|Kj is inside a component of dom(Gn) and that all iterates f(Kj),…,Gjn(Kj)∋y are outside In+1∪Jn+1. Since Gj+1n(x)=Gn(y), we obtain by induction that ⑤ holds for Gn+1 using that it holds for Gn and y instead of x.
The induced map G is defined as follows: for each n⩾0, each component of the domain J of Gn other than the central domains In+1 and Jn+1 becomes a component of the domain of G, and G|J=Gn|J.
Moreover, we compute by induction that if x∈(In∖In+1)∪(Jn∖Jn+1), and G(x)=fs(x), then
s⩽t0⋅(k0+1)…(kn−2+1)⋅(kn−1+1),
where t0=min{i>0,fi(c),fi(d)∈I0∪J0}. Note that for f∈B, t0=2.
3.3
The measure of the induced map
Note that the assumptions give that there exists a constant B with the following property: if J is any branch of Gk and Gk(J)=In, then
Leb({x∈J;Gk(x)∈In+m})|J|⩽B|In+m||In|.
(1)
The same result holds when Gk(J)=Jn. This trivially includes the branch of G0, that is the identity. Note that B is a distortion constant, and B⩽2 for ϵ sufficiently small and n⩾n0. Therefore, we can assume that B√ϵ/(1−√ϵ)<1/3. Moreover, |In|⩽ϵn−k|Ik| for all n⩾k⩾n0.
We use the notation α(y)=n if y∈(In∖In+1)∪(Jn∖Jn+1).
Lemma 3.2. If J is a branch of Gk−1 such that Gk−1(J)=In+1, then
Leb(x∈J;α(Gk(x))⩾n+1)⩽16|J|,
(2)
provided n⩾n0; analogously for Gk−1(J)=Jn+1.
Proof. Let In+1=K0⊃K1⊃…⊃Kkn=In+2 be as in Subsection 3.2. For each 0⩽i⩽kn−1 with Ki≠Ki+1, there can be at most two branches inside Ki, symmetric with respect to the critical point c, which map onto In+1 or Jn+1. Let P⊂Ki∖Ki+1 be such a branch (if it exists). We claim that P is ξ(ϵ)-well inside Ki. To see this, let s∈N be such that G|P=fs|P. We may assume that G(P)=In+1. In particular, G|P=Gi+1n|Ki. Then by our construction, fs−1 maps an interval T∋f(c) onto some interval Ij with j⩽n, and f−1(T)=Ki. Since In+1 is δ(ϵ)-well inside Ij, by Proposition 3.2, f(P) is ξ′(ϵ)-well inside T. Then the claim follows from the non-flatness of the critical point. Moreover, ξ(ϵ)→∞ as ϵ→0.
Let Un+1 be the union of those domains of G inside In+1∖In+2 that are mapped onto In+1 or Jn+1 by G. Then by the Koebe principle,
Leb({x∈J;Gk−1(x)∈Un+1})⩽110|J|.
It remains to consider branches J′ of Gk|J for which Gk(J′)=In′orJn′ with n′⩽n. Then, using the remark before this lemma, we have
Leb({x∈J′;Gk(x)∈In+1})|J′|⩽B|In+1||In′|⩽Bϵ⩽16.
This finishes the proof.
3.4
Acip for G
In this subsection we prove the existence of an acip for the induced map G.
Proof of Proposition 3.1. We will use the result given by Ref. [13] claiming that G has an acip if and only if there exists some η∈(0,1) and δ>0 such that for every measurable set A of measure Leb(A)<δ holds Leb(G−k(A))⩽η.
Write yn,k=Leb({x∈I0∪J0;α(Gk(x))=n}). Take C0>6Bmin{|In0|,|Jn0|}⋅(1ϵ)n0. We prove by induction that yn,k⩽C0⋅(√ϵ)n for all n,k⩾0. From the choice of C0, for all n⩽n0 and all k, we have
yn,k⩽1⩽C0⋅ϵn⩽C0⋅(√ϵ)n.
For k=0, it suffices to prove for n⩾n0+1. Indeed,
yn,0⩽(|In0|+|Jn0|)⋅ϵn−n0⩽ϵn⋅1ϵn0⩽C0⋅ϵn⩽C0⋅(√ϵ)n.
Now for the inductive step, assume that yn,k−1⩽C0⋅(√ϵ)n for all n. Pick n⩾n0+1. Write yn,n′,k−1=Leb({x∈I0∪J0;α(Gk−1(x))=n′andα(Gk(x))=n}). Therefore we have
If an acip ν exists, then it can be written as ν(A)=limn→∞1nn−1∑i=0Leb(G−iA). Therefore,
ν(In)+ν(Jn)⩽C0⋅(√ϵ)n.
Now take η∈(0,1). Fix n1 such that ∑n⩾n1yn,k<η/2 for all k>0. We need to show that we can choose δ>0 so that if A⊂I0∪J0 is a set of measure Leb(A)<δ, then Leb(G−k(A))<η for all k⩾0. By the choice of n1, it suffices to show that Leb(G−k(A))<η/2 for any A⊂(I0∖In1)∪(J0∖Jn1) and all k⩾0.
Without loss of generality, assume that A⊂In∖In+1 for some n<n1. Consider any branch Gk:J→In.
Case 1. If α(Gi(J))⩽n0 for all 0⩽i⩽k−1. By Mañé’s theorem[10], there exists C1=C1(f) such that D(Gk;J)⩽C1.
Case 2. If Gk|J can be extended to Gk:ˆJ→In−1 with n⩾n0, then by Proposition 3.2, D(Gk;J)⩽Kϵ.
In either case we have
Leb(G−kA∩J)⩽C′|A||In||J|⩽C′|A||In1||J|.
Case 3. There exist m⩾n and i⩽k that are maximal so that
m=α(GiJ)>α(Gi+1J)>…>α(Gk−1J)>n.
By Ref. [1, Proposition 2], any onto branch Gk:J→In can be written as ψ∘φ with
D(ψ)⩽logC2andφ∈F12(m−n+1).
Clearly, i⩾k−m+n−1. For such a branch, by Lemma 3.1,
Leb(G−kA∩J)⩽C2B(|A||In|)1/22(m−n+1)|J|.
For fixed m, the total measure of the set of points arriving at In in this fashion is bounded by k−1∑i=k−m+n−1ym,i⩽(m−n+1)C0⋅(√ϵ)m.
Thus Leb(G−kA)⩽η/4n1 for any k⩾0 and any A⊂In∖In+1, n<n1, with |A|<δ, provided δ is sufficiently small. It follows that if A⊂(I0∖In1)∪(J0∖Jn1) with Leb(A)<δ, then Leb(G−kA)<2n1η/(4n1)=η/2. This finishes the proof.
3.5
Acip for f
In this subsection we prove the main theorem.
Proof of Theorem 1.2. Let f∈B. Fix r⩾2. Let G be the Markov induced map of f. By Proposition 3.1, G admits an acip ν. Now it sufficed to show that we can pullback ν to obtain an acip for f.
Let In+1=K0⊃K1⊃⋯⊃Kkn=In+2 be as in Subsection 3.2. Since t<r, we can prove by induction that In+11⊂Kt∖Kt+1 and G|In+11 is exactly the first return onto In or Jn. The same holds for Jn+11. Now it is easy to check that kn=r. Therefore, if x∈(In∖In+1)∪(Jn∖Jn+1) and G(x)=fs(x), then
s⩽2(r+1)n.
Summing over all branches Jj⊂(In∖In+1)∪(Jn∖Jn+1), let sj denote the induced time on Jj. Then we find the partial sum
∑Jjsjν(Jj)⩽2(r+1)n(ν(In)+ν(Jn))⩽2C0(r+1)n(√ϵ)n
is exponentially small provided ϵ is sufficiently small.
Now this proposition follows by a standard pullback construction. Define μ by
μ(A)=∑isi−1∑j=0ν(f−jA∩Ji).
As f is non-singular with respect to Lebesgue, μ is absolutely continuous, and the f-invariance of μ is a standard result. The finiteness of μ follows directly from the fact that ∑siν(Ji)<∞.
4.
Conclusions
This paper considers the metric properties of a class of non-renormalizable cubic polynomials with generalized Fibonacci combinatorics. The interval maps with Fibonacci combinatorics are the candidate that has wild Cantor attractor ([8]) and have been studied widely for unimodal maps ([4][6]). The situation for multimodal is much more complicated. In this paper, we consider a special class of bimodal maps. The combinatorics of such a class were defined in terms of generalized renormalization based on the twin principal nest. The main result states that maps in this class with bounded combinatorics will have an absolutely continuous invariant measure. Moreover, we prove that any maps from such a class admits no Cantor attractor.
Conflict of interest
The author declares that she has no conflict of interest.
Bruin H, Shen W, van Strien S. Invariant measure exists without a growth condition. Communications in Mathematical Physics,2003, 241: 287–306. doi: 10.1007/s00220-003-0928-z
[2]
Graczyk J, Swiatiek G. The Real Fatou Conjecture. Princeton, USA: Princeton University Press, 1998.
[3]
Jakobson M, Swiatiek G. Metric properties of non-renormalizable S-unimodal maps. Part I. Induced expansion and invariant measures. Ergodic Theory and Dynamical Systems,1994, 14: 721–755. doi: 10.1017/S0143385700008130
[4]
Ji H, Li S. On the combinatorics of Fibonacci-like non-renormalizable maps. Commun. Math. Stat.,2020, 8: 473–496. doi: 10.1007/s40304-020-00210-x
[5]
Ji H, Ma W. Decay of geometry for a class of cubic polynomials. arXiv: 2304.10689, 2023.
[6]
Keller G, Nowicki T. Fibonacci maps re(al)-visited. Ergodic Theory and Dynamical Systems,1995, 15: 99–120. doi: 10.1017/S0143385700008269
[7]
Kozlovski O, Shen W, van Strien S. Rigidity for real polynomials. Annals of Mathematics,2007, 165 (3): 749–841. doi: 10.4007/annals.2007.165.749
Lyubich M. Combinatorics, geometry and attractors of quasi-quadratic maps. Annals of Mathematics,1994, 140: 345–404. doi: 10.2307/2118604
[10]
Mañé R. Hyperbolicity, sinks and measure in one-dimensional dynamics. Communications in Mathematical Physics,1985, 100: 495–524. doi: 10.1007/BF01217727
[11]
de Melo W, van Strien S. One-Dimensional Dynamics. Berlin: Springer-Verlag, 1993.
[12]
Shen W. Decay of geometry for unimodal maps: An elementary proof. Annals of Mathematics,2006, 163: 383–404. doi: 10.4007/annals.2006.163.383
[13]
Straube E. On the existence of invariant absolutely continuous measures. Communications in Mathematical Physics,1981, 81: 27–30. doi: 10.1007/BF01941798
[14]
Swiatek G, Vargas E. Decay of geometry in the cubic family. Ergodic Theory and Dynamical Systems,1998, 18: 1311–1329. doi: 10.1017/S0143385798117558
[15]
Vargas E. Fibonacci bimodal maps. Discrete and Continuous Dynamical Systems,2008, 22 (3): 807–815. doi: 10.3934/dcds.2008.22.807
Bruin H, Shen W, van Strien S. Invariant measure exists without a growth condition. Communications in Mathematical Physics,2003, 241: 287–306. doi: 10.1007/s00220-003-0928-z
[2]
Graczyk J, Swiatiek G. The Real Fatou Conjecture. Princeton, USA: Princeton University Press, 1998.
[3]
Jakobson M, Swiatiek G. Metric properties of non-renormalizable S-unimodal maps. Part I. Induced expansion and invariant measures. Ergodic Theory and Dynamical Systems,1994, 14: 721–755. doi: 10.1017/S0143385700008130
[4]
Ji H, Li S. On the combinatorics of Fibonacci-like non-renormalizable maps. Commun. Math. Stat.,2020, 8: 473–496. doi: 10.1007/s40304-020-00210-x
[5]
Ji H, Ma W. Decay of geometry for a class of cubic polynomials. arXiv: 2304.10689, 2023.
[6]
Keller G, Nowicki T. Fibonacci maps re(al)-visited. Ergodic Theory and Dynamical Systems,1995, 15: 99–120. doi: 10.1017/S0143385700008269
[7]
Kozlovski O, Shen W, van Strien S. Rigidity for real polynomials. Annals of Mathematics,2007, 165 (3): 749–841. doi: 10.4007/annals.2007.165.749
Lyubich M. Combinatorics, geometry and attractors of quasi-quadratic maps. Annals of Mathematics,1994, 140: 345–404. doi: 10.2307/2118604
[10]
Mañé R. Hyperbolicity, sinks and measure in one-dimensional dynamics. Communications in Mathematical Physics,1985, 100: 495–524. doi: 10.1007/BF01217727
[11]
de Melo W, van Strien S. One-Dimensional Dynamics. Berlin: Springer-Verlag, 1993.
[12]
Shen W. Decay of geometry for unimodal maps: An elementary proof. Annals of Mathematics,2006, 163: 383–404. doi: 10.4007/annals.2006.163.383
[13]
Straube E. On the existence of invariant absolutely continuous measures. Communications in Mathematical Physics,1981, 81: 27–30. doi: 10.1007/BF01941798
[14]
Swiatek G, Vargas E. Decay of geometry in the cubic family. Ergodic Theory and Dynamical Systems,1998, 18: 1311–1329. doi: 10.1017/S0143385798117558
[15]
Vargas E. Fibonacci bimodal maps. Discrete and Continuous Dynamical Systems,2008, 22 (3): 807–815. doi: 10.3934/dcds.2008.22.807