A NOTE ABOUT PRUNING AND H ´ENON MAPS
Abstract. We prove that there exists aC1-open setUof diffeomorphisms of the plane that consists of pruning diffeomorphisms of the horseshoe (up to a conjugacy), with the property that theC0-closure ofU contains all H´enon map in the boundary of the real horseshoe locus.
1. Introduction
In this note we study the H´enon family using pruning diffeomorphisms of the horseshoe.
First we will explain the pruning theory and then how it can be applied to understand C0-perturbations of the H´enon family.
1.1. Pruning theory. Letf0 be a diffeomorphism of the plane.
Definition 1.1. Let D be a simply connected domain of the plane and let P be its saturation ∪i∈Zf0i(D). An isotopyft:f0 'f1 is called apruning isotopy if:
(1) Supp(ft)⊂P, and (2) NW(f1) = NW(f0)\P,
where NW(f) denotes the non-wandering set of f.
Given a diskD, if there exists a pruning isotopy then we say that D is apruning disk.
The homeomorphismf1, the end point of the isotopy, is called apruning homeomorphism of f0. In our case Supp(ft) is the open set
(1.1) {p∈R2|there existst withft(p)6=f0(p)}.
When is a disk D a pruning disk? In [2] conditions are given to ensure that D is a pruning disk. We summarize these conditions in the following definition.
Definition 1.2. Let D be a disk with ∂D = γ1∪γ2 where γ1 and γ2 are arcs. We say that Dsatisfies the pruning conditions for f0 if
f0n(γ1)∩D=f0−n(γ2)∩D=∅,∀n≥1, and
n→∞lim diam(f0n(γ1)) = lim
n→∞diam(f0−n(γ2)) = 0.
In [2] de Carvalho proved the following result.
Theorem 1.3. If D satisfies the pruning conditions for f0 thenD is a pruning disk.
If we gather all the pruning homeomorphisms of a diffeomorphismf0then we will obtain a family ofmodels P(f0) such that we know how is exactly the dynamics of every element inP(f0): everyg∈ P(f0) has trivial dynamics in the saturation of a pruning diskD, and, outside ofD,g has the same dynamics as the initial diffeomorphism f0.
In this work we consider pruning homeomorphisms of a Smale horseshoe F, that is, f0 = F in the previous notation. F is a diffeomorphism defined in a region T formed by a square Q and 2 semi-circles. The action of F on T is as in Figure (1(a)). It is
2010Mathematics Subject Classification. Primary .
Key words and phrases. Pruning diffeotopies, H´enon maps, horseshoe locus.
1
2 A NOTE ABOUT PRUNING AND H ´ENON MAPS
well-known that, under certain conditions, F restricted to its non-wandering set NW(F) is hyperbolic and topologically conjugate to the shift σ defined in the space of symbol sequences Σ2 ={0,1}Z. See [7].
LetDN,M be the disk that contains the blocks corresponding to the symbolic sequences 0N1·010M and0N1·110M, forN, M ∈N. See Figure (1(b)). This diskDN,M is bounded by two segmentsγs, γupassing through the homoclinic pointsp0 =0∞110N−11·110M−1110∞ and p1 =0∞110N−11·010M−1110∞, such that γs ⊂Ws(0∞) and γu ⊂Wu(0∞). In [6]
we proved that, ifN, M >2 thenDN,M is a pruning disk for F. LetfN,M be the pruning homeomorphism associated to DN,M. Since there exist two open region L, E ⊂ DN,M such that γs∈∂Land γu ∈∂E that satisfy
(1.2) Fn(L)∩DN,M =F−n(E)∩DN,M =∅,∀n≥1, we can show the following:
Proposition 1.1. fN,M can be constructed as a hyperbolic diffeomorphism, when it is restricted to its non-wandering set.
This can be proved by making a C∞-isotopy St to the identity, supported in DN,M, such that S1 carries points of E to points in L. Thus the pruning isotopy isft=F ◦St. See [3] for results about differentiable pruning.
(a) The horseshoe mapF. (b) A pruning disk.
Figure 1. The action of the horseshoeF and a homoclinic pruning disk.
1.2. H´enon maps. We want to study perturbations of the H´enon maps (1.3) Ha,b(x, y) = (a−x2−by, x).
This is a family of diffeomorphisms defined in R2. Devaney and Nitecki [4] proved that if a >(5 + 2√
5)(1 +|b|)2 then Ha,b is a Smale horseshoe.
Let H be the set of parameter values (a, b) ∈R2 such that Ha,b is a Smale horseshoe.
This set is called the real horseshoe locus. See Figure (2).
Our result discusses perturbations of H´enon maps which are in ∂H. By Bedford and Smillie [1], if a H´enon map Ha0,b0 ∈ ∂H then it is a Smale horseshoe with an orbit of quadratic homoclinic tangencies between the stable and unstable manifolds associated to the fixed point 0∞. By Hoensch [5] and [1], this tangency occurs in the orbit of the
A NOTE ABOUT PRUNING AND H ´ENON MAPS 3
Figure 2. Horseshoe locus H.
homoclinic point t:=0∞1·010∞=0∞1·010∞ of the canonical coding. See Figure 3(a) for an example.
(a) H´enon map with a tangencyt. (b) Zoom inf0(t).
Figure 3. Invariant manifolds of the H´enon mapH3.3715,0.5∈∂H.
What can we say if we take small perturbations of H´enon maps that are in∂U? We claim that, up to conjugacy, we can find hyperbolic pruning diffeomorphisms of the horseshoe.
Actually we will prove the following theorem.
Theorem 1.4. There is a non-empty open set U ⊂Diff1(R2) such that (a) givenHa0,b0 ∈∂Hand >0, there existsg ∈ U satisfying
dC0(Ha0,b0, g)< ;
(b) if g ∈ U then g is hyperbolic and topologically conjugate to a pruning diffeomor- phism of the horseshoe, when both are restricted to their non-wandering sets.
2. Proof of Theorem 1.4
Letf0 :=Ha0,b0 ∈∂Hbe a H´enon map. Then Ha0,b0 has a homoclinic tangency at the point t(See [1, Theorem 2] and [5]).
Let K be the non-wandering set of Ha0,b0. By [1, Theorem 5.2], if T := Orb(t) de- notes the set of homoclinic tangencies between Wu(K) and Ws(K) then K is totally disconnected in a neighborhood of each point of K\T. Hence it is possible to construct homoclinic disks passing through homoclinic orbits.
4 A NOTE ABOUT PRUNING AND H ´ENON MAPS
Given >0, we must show that there are pointsp0 andp1, homoclinic to0∞, and near the tangency t, which define a pruning disk Dsuch that diam(f0(D))< .
We note that the pruning conditions are not modified if there exists a tangency in t between the stable and unstable manifolds of0∞. Let us take N, M ∈Nsufficiently large so that the pointsp0 :=0∞110N1·110M110∞andp1 :=0∞110N1·010M110∞are so close to the tangency that they define a pruning diskDN,M which satisfies diam(f0(DN,M))< . Since by [1, Theorem 5.2], every point ofK\Thas local product structure, we can prove thatf0is hyperbolic inK\P. Hence we can use the same construction of aC∞-isotopy as in Proposition 1.1 to construct a hyperbolic pruning diffeomorphismf1, associated toDN,M. Hence distC0(f1, f0)< . By hyperbolicity there exists an open setUa0,b0,N,M ∈Diff1(R2) such that every g ∈ Ua0,b0,N,M is topologically conjugate to the pruning diffeomorphism f1, when both are restricted to their non-wandering sets.
Let Ua0,b0 be the set
(2.1) Ua0,b0 = [
N,M >2
Ua0,b0,N,M.
The setU of the statement of Theorem 1.4 is U := [
(a0,b0)∈∂H
Ua0,b0.
3. Open question
When we study numerically the H´enon maps we observe that, if we change the parame- ters, many pieces of invariant manifolds (stable and unstable) are uncrossed. In particular if we take parameter values close to the boundary of H we have observed that for many of these parameters, the structures of the invariant manifolds are similar to the structures of the invariant manifolds of a pruning diffeomorphism in U. So it can be that the set U intersects the H´enon family in some domains of the parameter space. Hence the open question here is: how does U intersect the H´enon family?
References
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of Math (2).160(1), 1–26 (2004).
[2] de Carvalho, A.: Pruning fronts and the formation of horseshoes. Ergodic Theory and Dynam. Systems.
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[3] de Carvalho, A. and Mendoza, V.: Differentiable pruning and the hyperbolic pruning front conjecture.
Pre-Print, (2012).
[4] Devaney, R. and Nitecki , Z..: Shift automorphisms in the H´enon family. Comm. Math. Phys.67(2), 137–146 (1979).
[5] Hoensch, U.: Some hyperbolicity results for H´enon-like diffeomorphisms. Nonlinearity.21(3), 587–611 (2008).
[6] Mendoza, V.: Proof of the pruning front conjecture for certain H´enon parameters.
ArXiv:math.DS/1112.0705v1, (2011).
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