Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets
It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of . A renormalization approach has been used in [?] and [?] in a computer-assisted proof of existence of a “universal” area-preserving map — a map with orbits of all binary periods . In this paper, we consider maps in some neighbourhood of and study their dynamics.
We first demonstrate that the map admits a “bi-infinite heteroclinic tangle”: a sequence of periodic points , ,
whose stable and unstable manifolds intersect transversally; and, for any , a compact invariant set on which is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of symbols. A corollary of these results is the existence of unbounded and oscillating orbits.
We also show that the third iterate for all maps close to admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set:
ams:37E20, 37F25, 37D05, 37D20, 37C29, 37A05, 37G15, 37M99
Following the pioneering discovery of the Feigenbaum-Coullet-Tresser period doubling universality in unimodal maps [?], [?], [?] universality has been demonstrated to be a rather generic phenomenon in dynamics.
To prove universality one usually introduces a renormalization operator on a functional space, and demonstrates that this operator has a hyperbolic fixed point.
Such renormalization approach to universality has been very successful in one-dimensional dynamics, and has led to explanation of universality in unimodal maps [?], [?],[?], critical circle maps [?, ?], [?], [?] and holomorphic maps with a Siegel disk [?], [?], [?].
Universality has been abundantly observed in higher dimensions, in particular, in two and more dimensional dissipative systems (cf. [?], [?]), in area-preserving maps, both as the period-doubling universality [?], [?], [?], [?], [?], [?], [?] and as the universality associated with the break-up of invariant surfaces [?], [?], [?], [?], and in Hamiltonian flows [?],[?], [?], [?], [?], [?], [?], [?], [?] .
It has been established that the universal behaviour in dissipative and conservative higher dimensional systems is fundamentally different. The case of the dissipative systems is often reducible to the one-dimensional Feigenbaum-Coullet-Tresser universality ([?], [?], [?]). Universality for highly dissipative Hénon like maps
where is a unimodal map, - sufficiently small together with its derivatives, has been demonstrated in [?]. Specifically, it has been shown that these maps are in the Feigenbaum-Coullet-Tresser universality class, and that the degenerate map
where is the Feigenbaum-Coullet-Tresser fixed point, is a renormalization fixed point in that class. The authors of [?] have also constructed a Cantor set (an attractor) for infinitely renormalizable maps, and demonstrated that for such Hénon-like maps universality coexists with non-rigidity: the Hölder exponent of the conjugacy between the actions of any two infinitely renormalizable maps with nonequal average Jacobians on their Cantor sets has an upper bound less than .
The case of area-preserving maps seems to be very different, and at present there is no deep understanding of universality in conservative systems, other than in the “trivial” case of the universality for systems “near integrability” [?], [?], [?], [?], [?].
An infinite period-doubling cascade in families of area-preserving maps was observed by several authors in early 80’s [?], [?], [?], [?], [?]. The period-doubling phenomenon can be illustrated with the area-preserving Hénon family (cf. [?]) :
Maps have a fixed point which is stable for . When this fixed point becomes unstable, at the same time an orbit of period two is born with , . This orbit, in turn, becomes unstable at , giving birth to a period stable orbit. Generally, there exists a sequence of parameter values , at which the orbit of period turns unstable, while at the same time a stable orbit of period is born. The parameter values accumulate on some . The crucial observation is that the accumulation rate
is universal for a large class of families, not necessarily Hénon.
Furthermore, the periodic orbits scale asymptotically with two scaling parameters
To explain how orbits scale with and we will follow [?]. Consider an interval of parameter values in a “typical” family . For any value the map possesses a stable periodic orbit of period . We fix some within the interval in some consistent way; for instance, by requiring that the restriction of to a neighbourhood of a stable periodic point in the -periodic orbit is conjugate, via a diffeomorphism , to a rotation with some fixed rotation number . Let be some unstable periodic point in the -periodic orbit, and let be the further of the two stable -periodic points that bifurcated from . Denote with , the distance between and . The new elliptic point is surrounded by invariant ellipses; let be the distance between and in the direction of the minor semi-axis of an invariant ellipse surrounding , see Figure 1. Then,
where is the ratio of the smaller and larger eigenvalues of .
This universality can be explained rigorously if one shows that the renormalization operator
where is some -dependent coordinate transformation, has a fixed point, and the derivative of this operator is hyperbolic at this fixed point.
It has been argued in [?] that is a diagonal linear transformation. Furthermore, such has been used in [?] and [?] in a computer assisted proof of existence of a reversible renormalization fixed point and hyperbolicity of the operator .
An exploration of a possible analytic machinery has been undertaken in [?] where it has been demonstrated that the fixed point is very close, in some appropriate sense, to an area-preserving Hénon-like map
where solves the following one-dimensional problem of non-Feigenbaum type:
In this paper we will study the dynamics of the renormalization fixed point and maps in some neighbourhood of . We would like to emphasize that the preservation of area for such maps leads to one important difference between the case at hand and the highly dissipative case of Hénon-like maps: the former maps have no obvious invariant subsets in their domain. In particular, this means, that the construction of invariant Cantor sets carried out in [?], which is using existence of invariant subsets in a crucial way, is not readily applicable to the case of area-preserving maps.
To construct hyperbolic sets, we will use the idea of covering relations (see, e.g. [?, ?]) in rigorous computations. The Hausdorff dimension of the hyperbolic sets will be estimated with the help of the Duarte Distortion Theorem (see, e.g. [?]) which enables one to use the distortion of a Cantor set to ultimately find bounds on the dimension.
The structure of the paper is as follows: we begin by recalling the basic properties of area-preserving reversible maps in Section 2. In Section 3 we introduce some notation and recall some standard definitions from hyperbolic dynamics. In Section 4 we study the domain of analyticity of maps in a neighbourhood of the renormalization fixed point, and analytic continuation of the renormalization fixed point. Section 5 consists of the statements of our main theorems. In Section 6, we recall the definition and the main properties of the covering relations. In Section 7 we prove that any area-preserving reversible map in a neighbourhood of the renormalization fixed point has a transversal homoclinic orbit in its domain of analyticity. In Section 8 we construct a heteroclinic tangle for the renormalization fixed point. From its existence the existence of unbounded, and oscillating trajectories follow. In Section 9 we recall the Duarte Distortion Theorem. In Sections 10 and 11 we pr ove that the third iterate of any area-preserving reversible map in a neighbourhood of the renormalization fixed point has a horseshoe, and compute bounds on its Hausdorff dimension using the Duarte distortion theorem.
In a satellite paper [?], we prove that infinitely renormalizable maps in the neighbourhood of existence of the hyperbolic set for the third iterate also admit a “stable” set. This set is a bounded invariant set, such that the maximal Lyapunov exponent for the third iterate is zero. In [?] we provide an upper bound on the Hausdorff dimension of the stable set, and prove that the Hausdorff dimension is constant for all maps in some subset of infinitely renormalizable maps.
2 Renormalization for area-preserving reversible maps
An “area-preserving map” will mean an exact symplectic diffeomorphism of a subset of onto its image.
Recall, that an area-preserving map can be uniquely specified by its generating function :
Furthermore, we will assume that is reversible, that is
For such maps it follows from that
It is this “little” that will be referred to below as “the generating function”. It follows from (9) that is symmetric. If the equation has a unique differentiable solution , then the derivative of such a map is given by the following formula:
We will now derive an equation for the generating function of the renormalized map .
Applying a reversible twice we get
It has been argued in [?] that
We therefore set , to obtain:
If the solution of is unique, then , and it follows from that the generating function of the renormalized is given by
One can fix a set of normalization conditions for and which serve to determine scalings and as functions of . For example, the normalization
is reproduced for as long as
In particular, this implies that
Furthermore, the condition
is reproduced as long as
We will now summarize the above discussion in the following definition of the renormalization operator acting on generating functions originally due to the authors of [?] and [?]:
The Banach space of functions , analytic on a bi-disk
for which the norm
is finite, will be referred to as . will denote its symmetric subspace .
As we have already mentioned, the following has been proved with the help of a computer in [?] and [?]:
There exist a polynomial and a ball , , , such that the operator is well-defined, analytic and compact on .
Furthermore, its derivative has exactly two eigenvalues and of modulus larger than , while
Finally, there is an such that
The scalings and corresponding to the fixed point satisfy
The radius of the contracting part of the spectrum has been estimated in [?] to be .
It follows from the above theorem that there exist a codimension local stable manifold .
A reversible map of the form (10) such that is called infinitely renormalizable. The set of all reversible infinitely renormalizable maps is denoted by .
3 Some notation and definitions
We will use the following notation for the sup norm of a function and a transformation defined on some set :
where and are projections on the corresponding components.
We will also use the notation for the norm for vectors in .
The interval enclosures of and will be denoted
The corresponding interval enclosure for the linear map will be denoted ; if , then
The bound on the fixed point generating function will be called :
while the bound on the renormalization fixed point will be referred to as :
where is as in Theorem 2.3; the third iterate of this bound will be referred to as . With we denote an interval matrix valued function such that
where is the domain of , and the bound on the operator norm of for , on a set will be denoted
Given a non-empty open set we will denote by the set of reversible area-preserving maps , analytic on .
We will proceed with a collection of classical notations (see, eg, [?]) relevant to our following discussion.
(Hyperbolic set) Let be a smooth manifold, and let be a diffeomorphism of an open subset onto its image.
A set is called hyperbolic for the map if there is a Riemannian metric on a neighbourhood of , and , such that for any and the tangent space admits a decomposition in two invariant subspaces:
on which the sequence of differentials is hyperbolic:
(Locally maximal hyperbolic set) Let be a hyperbolic set for . If there is a neighbourhood of such that then is called locally maximal.
(Bernoulli shift) Let be the space of all two-sided sequences of symbols:
Define the Bernoulli shift on as
(Topological Markov chain) Let be an matrix whose entries are either or . Let
is called a topological Markov chain determined by .
(Homoclinic and heteroclinic points) Let be a homeomorphism on a metric space . A point is said to be homoclinic to the point if
A point is said to be heteroclinic to points and if
In this case we will also say that there exists a heteroclinic orbit between points and .
If is a differentiable manifold, and is a hyperbolic fixed point of , then we say that is a transversal homoclinic point to if it is a point of transversal intersection of the stable and unstable manifolds of .
Let be a metric space. If , and , the -dimensional Hausdorff content of is defined as
The Hausdorff dimension of is defined as
4 Domain of analyticity of and
is defined implicitly by the generating function , its domain is given as in [?]:
To find the domain of , we note that its second argument is equal to , for some (see ). Thus, the domain of , , is given by:
We denote by
the real slice of .
To solve nonlinear equations on the computer we use the interval Newton operator, see e.g. [?].
(Interval Newton Operator) Let , and let be an interval matrix valued function such that for all . Let be a Cartesian product of finite intervals, , and assume that if , then is non-singular. We define the interval Newton operator as:
The main properties of is that if , then there exists a unique solution to in , which is contained in , and if , then there is no solution to in .
There exists a non-empty open set ,
such that for every there exists a unique solution of the interval Newton operator of the function , , that satisfies .
Proof. Set . Given , let be the interval Newton operator (for some appropriately chosen ). We have verified that there exists a non-empty set , such that for all
This verification is implemented in the program findDomain of [?] It follows, see e.g. [?], that there is a unique , such
that is in the real slice of and . Thus, is defined as a function of , and by (31) .
The generating function is analytic on a bi-disk, preferably should have a similar property. From the above construction, we can at least show that is defined on a complex neighbourhood of from the above Lemma 4.2:
contains an open complex neighbourhood of the set .
is only well defined if , so that the
condition of the implicit function theorem is automatically satisfied on
the solution set of . Since
is analytic on the bi-disk , there is an open neighbourhood in of the
solution set where . It follows from the implicit
function theorem that is analytic on this neighbourhood.
Generally, we will denote the third iterate of a map as . The domain of , , is given by
Using the program findDomain, we have verified that the real slice of is an open non-empty set. Approximations of and are shown in Figure 2. We note that by using the renormalization equation , has, for any , an analytic continuation to domains
while has an analytic continuation to
The real slices of the domains and are given in the Figure 3.
We will conclude this Section with the following
All maps posses a hyperbolic fixed point , such that
, and ;
has two real eigenvalues:
The bound on the fixed point has been obtained with the help of the interval Newton method. Hyperbolicity has been demonstrated by computing a bound on using formula .
5 Statement of the main results
We will now summarize our main findings.
Our first theorem describes the two-sided heteroclinic tangle for the fixed point map .
Main Theorem 1
The renormalization fixed point has the following properties:
possesses a point which is transversally homoclinic to the fixed point ;
there exists a positive integer such that for any negative integer the map has a heteroclinic orbit between the periodic points and , and for any positive integer the map has a heteroclinic orbit between the periodic points and ;
for any and there exists an integer and an invariant set , such that
where is the topological Markov chain defined by a tridiagonal matrix
for any there exists a point and an such that (an unbounded orbit), and a point and such that and (an oscillating orbit).
Our second result demonstrates that all locally infinitely renormalizable maps admit a hyperbolic set in their domain of analyticity.
Main Theorem 2
Any admits a hyperbolic set for ;
whose Hausdorff dimension satisfies:
where is strictly positive.
6 Topological tools
The main tools of our proofs are covering relations [?, ?] and cone conditions [?, ?], see also [?] for proving the existence of homoclinic and heteroclinic orbits. To make the present paper reasonably self-contained we include a brief introduction to the necessary concepts.
6.1 H-sets and Covering relations
The notion of an h-set and a covering relation first appeared in [?], the most thorough treatment is [?]. The basic idea is to construct computable conditions for the existence of a semi-conjugacy to symbolic dynamics. This is done by constructing h-sets, i.e. hyperbolic-like sets, that cross each other in a (topologically) non-trivial way. We denote by , the open ball in with centre and radius , and .
An h-set is a quadruple consisting of
a compact subset of ,
a pair of numbers with ,
a homeomorphism , such that
We denote such a quadruple by . We usually drop the bars on the support and refer to it as . Furthermore,
and are the nominally unstable and stable directions, respectively. The idea of a covering relation between two h-sets is that the image of the first should be mapped transversally across (the exit set of ) and inside of (the entrance set of ). We formally define it.
Assume are h-sets, such that and . Let be a continuous map. Let . We say that
( covers ) iff the following conditions are satisfied
1. There exists a continuous homotopy , such that the following conditions hold true
2. There exists a linear map , such that
is called a model map for the relation .
The maps that we study in the present paper are reversible, which we use to reduce the amount of computation. For such maps the following definition is useful, see also the discussion in [?].
Let be an -set. We define the -set as follows:
The compact subset of the quadruple is the compact subset of the quadruple , also denoted by ,
The homeomorphism is defined by
where is given by .
Assume N,M are -sets, such that and . Let . Assume that is well defined and continuous. We say that ( backcovers ) iff .
Let and be -sets. We say that generically -covers () if or .
The main property of covering relations is contained in the following theorem [?, Corollary 7].
Assume that we have the following chain of covering relations:
then there exists a point , such that
Moreover, if , then can be chosen so that
In our proofs the hypothesis of Theorem 6.6 is verified using the routine checkCoveringRelations of [?].
6.2 Cone conditions for h-sets
Theorem 6.6 gives a computational tool to prove the existence of orbits with prescribed symbolic dynamics. To prove that such orbits are unique one would ideally require hyperbolicity of the map in a neighbourhood of the orbit. Typically, this is proved by constructing invariant cone fields. An alternative method to prove uniqueness is provided by covering relations with cone conditions, first described in [?], the method is studied in further detail in [?], which we follow below.
Let be an -set and be a quadratic form