Discretizations of a generic dissipative homeomorphism
4.2 Dynamics of discretizations of a generic homeomorphism
We now establish a discrete counterpart of the shredding lemma. Since having basins of attraction is stable by perturbation, we have a similar statement for discretiza-tions of a generic homeomorphism. In what follows these arguments are developed.
To each point xN ∈ EN, we associate a closed set PN−1({xN}), made of the points in X one of whose projections on EN isxN. The closed sets PN−1({xN}) form a basis of the topology of X when N runs through Nand xN runs through EN. Let f ∈ Homeo(X), N∈Nandϑ:N→R∗+be a function that tends to +∞at +∞. Letδ,ε>0 and Uj be the sets obtained by the shredding lemma forf,δandε.
For all j≤` we denote by UejN the union overxN ∈EN of the closed sets PN−1({xN}) whose intersection with Uj are non-trivial. Then eUjN tends to Uj for the metric d(A,B) =λ(A∆B), and for the Hausdorffmetric; the shredding lemma states that these convergences are independent from the choice ofε0. Thus, for allkbig enough, prop-erties (i) to (iii) of the shredding lemma remain true for the discretizations gN (for arbitraryg satisfying the properties of the lemma). Takingε0 small enough and modi-fying a littlegif necessary, there exists sets Wj,i andg∈Homeo(X) such thatd(f , g)<δ, Wj,i⊂PN−1({xN}) and Card(S
j,iWj,i∩EN)≤ϑ(N). The others estimations over the sizes of the sets involved in the lemma are obtained similarly. Finally, we obtain the following lemma:
Lemma 4.9 (Discrete shredding lemma). For all f ∈ Homeo(X), for all ε,δ >0 and all functionϑ:N→R∗+that tends to+∞at+∞, there exists integersN0,`and`1,· · ·, `` bigger than 1/ε, and a homeomorphism g ∈Homeo(X) such thatd(f , g)<δ and for all N≥ N0, there exists a family of subsetsU1N,· · ·,U`NofENsuch that:
(i) gN(UjN)⊂UjN, (ii) Card(UjN)<εqN, (iii) CardS`
j=1UjN
>(1−ε)qN,
(iv) Card(gN(UjN))<Card(UjN)/(`ϑ(N0)), (v) for allj, there exists subsetsWj,1N,· · ·,Wj,`N
j ofENsuch that a) Card(S
j,iWj,iN)≤qN/ϑ(N0),
b) gN(Wj,iN)⊂Wj,i+1N , for everyi∈ {1,· · ·, `j−1}andgN(Wj,`N
j)⊂Wj,1N,
c)
UjN⊂ [
m≥0
gN−m
`j
[
i=1
Wj,iN ,
(vi) for allj and all i, there exists sets Uj and Wj,i satisfying properties (i) to (v) of the shredding lemma such thatUj ⊂PN−1(UjN)andWj,i⊂PN−1(Wj,iN).
(vii) ifN≥N0, then for alljand alli, we have X
j
dH(Uj,PN−1(UjN))<ε and X
i,j
dH(Wj,i,PN−1(Wj,iN))<ε (wheredHis the Hausdorffmetric), and
X
j
λ(Uj∆PN−1(UjN))<ε and X
i,j
λ(Wj,i∆PN−1(Wj,iN))<ε.
Remark4.10. Properties(i)to(v)are discrete counterparts of properties(i)to(v)of the continuous shredding lemma, but the properties(vi)and (vii)reflect the convergence of the dynamics of discretizations to that of the original homeomorphism.
This lemma implies that we can theoretically deduce the behaviour of a generic homeomorphism from the dynamics of its discretizations. The next section details this remark.
To begin with, we deduce from the shredding lemma that the dynamics of discretiza-tionsfNtends to that of the homeomorphismf. More precisely almost all orbits of the homeomorphism areδ-shadowedby the orbits of the corresponding discretizations.
Definition 4.11. Let f and g be two maps from a metric space X into itself, x, y ∈X andδ>0. We say that the orbit ofx byf δ-shadowsthe orbit ofy byg if for allm∈N, d(fm(x), gm(y))<δ.
Corollary 4.12. For a generic homeomorphism f ∈Homeo(X), for allε>0and allδ>0, there exists an open setAsuch thatλ(A)>1−εandN0∈N, such that for allN≥N0and all x∈A, the orbit ofxN= PN(x)byfNδ-shadows the orbit ofxbyf.
Therefore, for a generic homeomorphismf, there exists a full measure dense open setO such that for allx∈O, allδ>0and allNlarge enough, the orbit ofxNbyfNδ-shadows that ofxbyf.
Proof of Corollary4.12. This easily follows from the discrete shredding lemma, and es-pecially from the fact that the sets Wj,iN tend to the sets Wj,i for Hausdorff metric, in particular O =S
j,εUj,ε.
This statement is a bit different from the genericity of shadowing (see [PP99]): here the starting point is not a pseudo-orbit but a pointx∈X; Corollary4.12expresses that we can “see” the dynamics off on that offN, with arbitrarily high precision, provided that N is large enough. Among other things, this allows us to observe the basins of attraction of the neighbourhoods of the Lyapunov stable periodic points of f on dis-cretizations. Better yet, to each family of attractors (Wj,i)i of the basin Uj of the home-omorphism corresponds a unique family of sets (Wj,iN)i that are permuted cyclically by fN and attract a neighbourhood of Uj. Thus, attractors are shadowed by cyclic orbits
offN and we can detect the “period” of the attractor (i.e. the integer`j) on discretiza-tions: when N = N0, the sets Wj,iNeach contain only one point; a phenomenon of period multiplication might appear for N bigger [Bla89, Bla86, Bla84, Bla94, AHK03]. This behaviour is the opposite of what happens in the conservative case, where discretized orbits and true orbits are very different for most points.
Again, in order to show that the dynamics of discretizations converges to that of the initial homeomorphism, we establish the convergence of attractive sets offN to that of f. Recall that A0 is the closure of the set of Lyapunov stable periodic points off (see Definition4.3).
Proposition 4.13. For a generic homeomorphismf ∈Homeo(X), the recurrent setsΩ(fN) tend weakly to A0 in the following sense: for all ε >0, there exists N0 ∈ Nsuch that for all N≥ N0, there exists a subset eEN of EN, stabilized by fN, such that, notingΩ(fe N) the corresponding recurrent set, we have Card(eCard(EEN)
N) >1−εanddH(A0,Ωe(fN))<ε.
Proof of Proposition4.13. Letε>0. For all N∈N, leteENbe the union of the sets UjNof Lemma4.9for the parameterε. This lemma ensures thateEN is stable byfN and fills a proportion greater than 1−εof EN. We also denote byΩ(fe N) the associated recurrent set:
Ωe(fN) = [
x∈S` j=1UejN
ωf
N(x).
Property(viii)of Lemma4.9ensures that
Nlim→+∞dH(Aε,Ω(fe N))<ε.
To conclude, it suffices to apply Corollary4.5which asserts that Aε→A0for Hausdorff distance.
We now set a final consequence of Lemma4.9, which reflects that the ratio between the cardinality of the image of discretizations and which of the grid is smaller and smaller:
Corollary 4.14. Letϑ:R+→R∗+be a function that tends to+∞at+∞. Then for a generic homeomorphismf ∈Homeo(X),
lim
N→+∞
ϑ(N)Card(fN(EN)) Card(EN) = 0 ;
more precisely, for everyM∈N, there existsN0≥Msuch that for everyN≥N0, we have Card(fN(EN))
Card(EN) ≤ 1 ϑ(N0).
In particular, the degree of recurrence satisfieslimN→+∞D(fN) = 0.
Remark that asΩ(fN)⊂fN(EN), the same estimation holds for the recurrent set. This corollary can be seen as a discrete analogue of the fact that a generic homeomorphism is totally singular,i.e. that there exists a Borel set of full measure whose image underf is zero measure. Again, it reflects the regularity of the behaviour of the discretizations of a dissipative homeomorphism: generically, the behaviour of all (sufficiently fine)
discretizations is the same as the physical behaviour of the initial homeomorphism.
This is very different from the conservative case, where sometimes fN(EN) = EN and sometimes Card(fN(EN))≤ϑ(N) whereϑ:R+→R∗+is a given map that tends to +∞at +∞.
The discrete shredding lemma also allows us to have an estimation about the com-binatorial behaviour offNΩ(fN):fNhas a lot of periodic orbits, and along these periodic orbits a lot have long lengths.
Corollary 4.15. For a generic homeomorphismf ∈Homeo(X)and for every M∈N, there existsN0∈Nsuch that for everyN≥N0, there exists a subseteEN of EN, stabilized byfN, such that we have Card(eCard(EEN)
N)>1−ε(as in Proposition4.13), such that all the periodic orbits of fN|eE
N have length bigger thanM, and such that the number of such orbits is bigger thanM.
It remains to study the behaviour of measuresµfUN (see Definition3.10). To do that, we have to suppose that the sequence of grids well behaves with respect to the measure λ. Again, the results are very different from the conservative case: for any open set U, the measuresµfUN tend to a single measure, sayµfU.
Theorem 4.16. For a generic homeomorphismf ∈Homeo(X)and an open subsetU ofX, the measureµfU is well defined7 and is supported by the setA0. Moreover the measuresµfUN tend weakly toµfU.
Sketch of proof of Theorem4.16. The proof of this theorem is based on the shredding lemma: the set of homeomorphisms which satisfy the conclusions of the lemma is a Gδdense, so it suffices to prove that such homeomorphismsf satisfy the conclusion of the proposition. Let U be an open subset of X andϕ: X→Rbe a continuous function.
We want to show that on the one hand the integralR
Xϕ dµfUis well defined,i.e.that the Birkhofflimits for the functionϕ
mlim→+∞
1 m
m−1
X
i=0
ϕ(fi(x))
are well defined for almost everyx∈U; and on the other hand we have the convergence Z
X
ϕdµfUN −→
N→+∞
Z
X
ϕ dµfU,
For the first step, the idea of the proof is that most of the points (forλ) eventually belong to a set Wj,i. Since the iterates of the sets Wj,i have small diameter, by uniform continuity, the function ϕ is almost constant on the sets fm(Wj,i). Thus the measure µfx is well defined and almost constant on the set of points whose iterates eventually belong to Wj,i. And by the same construction, since the dynamics offNconverge to that off, and in particular that the sets UjN and{wj,iN}converge to the sets Uj and Wj,i, the measuresµfUN tend to the measuresµfU.
7. In other words, a generic homeomorphism is weird, see Definition4.1, see also [AA13].