Statistical properties of coupled expanding maps on a lattice with general infinite range couplings and Hölder densities
We continue the development of transfer operator techniques for expanding maps on a lattice coupled by general interaction functions. We obtain a spectral gap for an appropriately defined transfer operator, and, as corollaries, the existence of an invariant conformal probability measure for the system, exponential decay of correlations, the central limit theorem and the almost sure invariance principle.
Key words and phrases:Coupled Lattice Maps, Statistical Properties of Dynamical Systems, Almost Sure Invariance Principle.
2000 Mathematics Subject Classification:37A05, 37A25, 37A60, 60G10, 60F05, 80C20
For one dimensional dynamical systems, the conditions under which there exists a unique ergodic invariant probability measure, supported on an invariant attractor and governing the dynamics of initial conditions in the basin of the attractor, are well understood. The complexity of the geometry in higher dimensions makes the problem much harder, however, in certain cases, significant results have been proven. For instance, the uniformly hyperbolic case has been presented in [MR2423393], the non-uniformly hyperbolic case of the Hénon map in [MR1087346, MR1218323], and recently, the dimensional analogue in [MR2415378]. In all, even though the 1 dimensional case is better understood in general than the dimensional setting for , the theory of SRB measures for finite dimensional dynamical systems is fairly complete (for an excellent, though somewhat dated, overview, see [MR1933431] and the references therein).
However, outside of a few settings, not much is known in the infinite dimensional setup. The primary problem is one of methodology, and the settings in which something can be said about an infinite dimensional dynamical system are those in which the techniques applicable in the finite dimensional case can be extended to infinite dimensions. Coupled maps on a lattice (CML) provide an example of infinite dimensional systems that can be studied by extensions of finite dimensional techniques, and for this reason, they have been extensively studied.
An excellent review of the theory of CML is provided in [MR1464237]. In recalling the background on CML theory, we will be more restrictive; in particular, we will focus only on the development of ideas relevant to the results in this paper. An imprecise definition of a CML is provided in this section to set the context of the results that we mention. A precise formulation follows in later sections. Suppose we have a map on and let . We have an extended map on the space which can be defined as where and Suppose now we have interactions between the different dynamical systems . In the simplest case of being the nearest neighbor diffusive coupling, we can specify the interactions as, for a given ,
therefore becomes a parameter that tunes the strength of the interactions between the nodes of the lattice. We note also that this coupling specified has range 1, because only the closest neighbors to each node influence the state of that node. The system under study is now iterations of .
Bunimovich and Sinaĭ [MR967468] considered as expanding maps of the interval with nearest neighbor diffusive coupling with a non-constant diffusion strength, chosen so that the coupling map was onto. They established that if the interval map exhibits sufficient expansion, then there exists a unique invariant measure, mixing in time and space. They construct the invariant measure as the limit of the Gibbs measures for the finite dimensional projections of their coupled system. Then, in [MR1184471] the authors implement a numerical algorithm to extract orbits periodic in time and space for a 1-d lattice of Hénon maps, coupled by a weak nearest neighbor diffusive coupling. They define a new family of Lyapunov exponents that estimate the growth rate of spatially inhomogeneous perturbations.
Following the results in [MR967468] and [MR1184471], a natural question arose: if the coupling is viewed as perturbation of the uncoupled system, what can happen if the coupling strength becomes large relative to the inherent stability of the uncoupled system? This question was studied in [MR1184470], where the authors considered CML, where the maps on each node were one dimensional with a globally attracting stable periodic trajectory. These maps are coupled by a diffusive nearest neighbor coupling. The authors prove that if the coupling strength is sufficient large, then the phase space of the coupled map lattice is split into a complicated partition with many basins of different attractors.
The question of phase transitions for a one parameter family of finite-dimensional CML, with the coupling strength as the parameter of interest, was studied further, theoretically and numerically, by [MR1192055]. They obtained sufficient conditions in terms of that the lattice have a continuum of ergodic components when the individual node maps are the doubling maps. They also produced an example of a function for which the uncoupled system is mixing, whereas for a suitable , there are several domains in the phase space, interchanging with the period 2.
Keller and Künzle [MR1176625] then investigated transfer operator techniques for CML maps. They also considered expanding maps on each node of the lattice, and coupled them by a weak coupling (small ). The authors established the spectral theory using spaces of bounded variation, closely following the setting in [MR967468]. Then, in a three-part paper, Gundlach and Rand [MR1211566, MR1211567, MR1211568] developed the stable manifold theory for coupled lattice maps with short or finite range interactions (Part I), and use this to establish the existence of a natural spatio-temporal measure that plays the role of the usual SRB measure in the case of temporal systems (Part III) and the existence and uniqueness of Gibbs states for higher dimensional symbolic systems by using thermodynamic formalism (Part II). The transfer operator approach of [MR1176625] was developed further by [MR1981170] where the author established a spectral gap for weakly coupled real-analytic circle maps. In [MR1936014], the author then modified the approach in [MR1981170] to construct generalized transfer operators associated to potentials and establish a spectral gap for small potentials for weakly coupled analytic maps. Some limit theorems, such as the central limit theorem, moderate deviations, and a partial large deviations result were also established.
Returning to the setting of expanding real maps, [MR2083421] gave a proof of the existence, uniqueness and exponential mixing of invariant measures for weakly coupled lattice maps without cluster expansion. The coupling considered was finite range and had a very specific form, and the transfer operator was considered on the space of measures of bounded variation with absolutely continuous finite dimensional marginals. Subsequently, efforts were made to admit more general couplings and more general maps on the lattice nodes, and in [MR2395729] the authors study a one-dimensional lattice of weakly coupled piecewise expanding maps of the interval. Strong assumptions are still required on the coupling, however, the authors do not require that the coupling be a homeomorphism of the infinite dimensional state space. They prove that the transfer operator defined on an appropriate space of densities with bounded variations, with absolutely continuous finite dimensional marginals (with respect to the Lebesgue measure), has a spectral gap. This implies that there exists a measure with exponential decay of correlations in time and space. In [MR2200881] the authors extend the results established in [MR2395729] to include lattices of any dimensions and couplings of infinite rage with the coupling strength decaying exponentially in space.
In the setting of [MR2083421] and [MR2200881] the Bardet, Gouëzel and Keller [MR2293068] prove the central limit theorem and the local limit theorem for Lipschitz functions depending on finitely many coordinates. The proof of the local limit theorem requires the additional assumption that for any compact interval
here is the variance in the central limit theorem. As in [MR2083421, MR2200881], the transfer operator is defined on the Banach space of measures of bounded variation with absolutely continuous finite dimensional marginals.
In this paper we continue the development of transfer operator techniques for studying existence and uniqueness of invariant measures corresponding to an initial potential in an appropriate class of potentials for expanding maps on a lattice coupled by an infinite range coupling. As in the setting of [MR2083421, MR2200881], we obtain a spectral gap for the transfer operator in an appropriate space of densities. In contrast to the setting of [MR2083421, MR2200881], we obtain our potentials and densities from a class of Hölder functions, and admit spatial couplings more general that those previously considered. We do not assume the existence of a reference measure; the theory is developed with respect to suitable potentials. Finally, we only require that the minimal expansion and the coupling strength be related as for the existence, and uniqueness, of the invariant probability measure. We then use an abstract result of Gouëzel to obtain the almost sure invariance principle for our system. We also show that the invariant probability measure is conformal on open sets.
2. Main results.
In this section, we will list the main theorems that we prove. The precise definitions for the objects that appear here will follow in the body of the paper.
We start with a result known in the dynamical systems literature as “decay of correlations”. We establish this result by showing that an appropriate transfer operator has a spectral gap. For more details on the operator see sections and 5. We also check that the Lasota-Yorke inequality is stable under perturbation and in doing so, establish the almost sure invariance principle by using [Go2010]. We use the space of Hölder continuous functions with the norm where is the -Hölder constant (the precise definition is below).
Let be the coupled lattice map on the lattice system and the space of Hölder continuous functions on .
Then for any potential function there exists an invariant measure which is -conformal where for an . Moreover there exists a constant with the property that for any there exists a constant such that
Finally, by using an abstract result from [Go2010], and Theorem 4.1, we prove that
The system satisfies the almost sure invariance principle.
Various statistical properties such as the law of iterated logarithms, the weak invariance principle and the central limit theorem now follow as corollaries.
(CLT) For observables one has
as where is the coupled lattice map and is the normal distribution.
A technical result along the way is the following proposition:
Let . Then and moreover there exists a constant and a constant such that
The proposition is used to prove the existence of the invariant probability measure which satisfies . This result is proved as Theorem 4.1. As a corollary to this theorem, with suitable modifications to the classical techniques we prove that the essential spectral radius of is strictly smaller than the spectral radius of (which is 1).
Finally, we make a note regarding constants. We denote ‘global’ constants by throughout the paper and ‘local’ constants by for each lemma or theorem. The constant tunes the strength of interactions between the maps on the lattice.
3. The uncoupled system.
First, we need to define what the admissible class of observables is. In order to study observables (and potentials) on an infinite lattice, we need to define how well approximated these observables are by restrictions to finite sub-lattices. Let and the lattice space where is the unit interval. Define
Assume the following:
3.1. Assumptions on the metric
Let be a metric on the unit interval and pick . Define a metric on by
where are points in .
3.2. Assumptions on the map on
The global map on is thought to be composed of individual maps on the nodes . We list below the conditions that our map satisfies (we note that these are not the most general conditions under which our theorem can be stated, but in order to keep our reasoning transparent, we do not pursue a more general setting).
The map has full branches. As a consequence, for each , is constant. This also implies that the map has at least one fixed point. Denote one such fixed point by
The map is expanding. This is taken to mean that there exists a constant such that for every inverse branch of one has ( are the images of under , i.e. ).
Define as and put for the “projection” which is given by where .
For we define the “restriction” to as
that is . A consequence of the fact that outside of the lattice we have chosen to be equal to is that
3.3. Definition of the function space
For we define the Hölder constant
defines a norm and we define the function space
We will also sometimes need the variation semi-norm for some given by
is the th variation of . If then for all . Therefore, in what follows, we fix some and, instead of writing , we only write .
3.4. Defining the positive operators and
Let be a potential function and define for the finite sub-lattice the transfer operator for by
, where the summation is over inverse branches of , i.e. is the identity and moreover the image of under has the property that for . For the normalising factor one has as .
Clearly, is a well defined, positive, bounded linear operator on functions on . The following lemma serves to define the transfer operator by taking a limit to infinity.
Let such that . Then
(I) is uniformly (in ) bounded.
(II) For such that the sequence is Cauchy for each .
(III) for some constant .
(I) Clearly , which implies is bounded uniformly in .
(II) For one has
where are inverse branches in and are inverse branches in .
The lattice contains elements more than , and so . Therefore the above sum simplifies as
Therefore (for some )
from where it follows that
for some constants as by part (I) is uniformly bounded.
(III) This follows from the first inequality in the last line of estimates. ∎
Since by Lemma 3.1, for each , the sequence is a Cauchy sequence (of real numbers) we now define the operator for the infinite lattice system as the pointwise limit:
is a non-negative and continuous operator on .
Clearly is non-negative as the approximations are non-negative. Since is a linear operator, it is enough to show continuity at the origin. For we see that is a bounded operator in the -norm as
Now let , then
for some as (). Hence for some constant which is independent of . ∎
Note that the space with the metric is convex, compact and separable. Separability follows from the fact that for every there are finitely many points in that are -dense in for any . In this way one produces a countable dense set in . This implies that the set of probability measures on is compact in the weak* topology. Thus following [MR2423393] we can define an operator by where . By the theorem of Schauder-Tychonoff has a fixed point in . Thus , where .
Let and put . Define the function set
Notice that .
For one has and which implies
where as the diameter of is equal to and (i.e. ). This implies that for some . Hence . ∎
In order to apply the theorem of Schauder-Tychonoff we must first show that the operator maps into itself.
maps into itself.
Clearly for all . Since is a positive operator we also get for all . It remains to verify the regularity property. Since
There exist a unique so that and moreover is strictly positive.
The set is convex and equicontinuous by Lemma 3.4. Therefore is compact in the -norm by Arzela-Ascoli and has by Schauder-Tychonoff a fixed point . That is . To see that is strictly positive assume that has a zero at , i.e. . Then
where the sum is over all inverse branches of in . Since this implies that for all inverse branches of . Since the set is dense in and is continuous we conclude that is identically zero which contradicts the assumption .
To obtain uniqueness of assume that there is a second eigenfunction so that and put . By convexity of one has and by choice of there exists an so that . By the argument above we conclude that must vanish identically, which is impossible. Hence is unique. ∎
We now define the normalized transfer operator by putting for . Note that is well defined since and has the potential function . Moreover has the (dominant) simple eigenvalue with eigenfunction as . The associated eigen-functional is a -invariant probability measure. Define, also, as follows:
where Note that by definition, , and
Hence, we state a corollary to Lemma 3.1:
For each , and for each , the sequence is Cauchy, and hence it converges to .
The fact that is Cauchy, and hence has a point wise limit, follows directly from Lemma 3.1. Also, notice that
The term in the last summation can be written as
and so, because , for any , we have that
Similarly, we obtain that This completes the proof. ∎
3.5. Lasota-Yorke inequality for
We now establish a Lasota-Yorke inequality for the operator . We do this by obtaining the corresponding inequality for each approximation . We use the following notation: For a function we denote by its th ergodic sum. We shall need the following technical estimate.
Let be a potential and let
Then for all and :
we get that
Note that is bounded by and is bounded by . Define . The above combined with
proves the desired bound. ∎
Proposition 3.9 (Lasota-Yorke inequality for ).
Let and be as before. Let . Then (depending on and ) such that
as . By Lemma 3.8, we have
where . Consequently
Now we can prove the ’Lasota-Yorke’ inequality for the operator for the uncoupled map on the full, infinite lattice .
Let and let be as before, and . There exists a constant depending only on such that
Recall from Assumption (2). Let . Then for all one has