Discretizations of a generic conservative homeomorphism
5.3 Individual behaviour of discretizations
Now we have set the technical Theorems5.2and 5.5, we can establish results con-cerning the behaviour of discretizations of a generic conservative homeomorphism.
Here we study individual behaviour of discretizations, i.e. properties about only one order of discretization. As has already been said, applying Theorem5.2, it suffices to find dense types of approximation to obtain properties about discretizations. In prac-tice, these dense types of approximations are obtained from variations of Lax’s theorem (Theorem5.5).
Recall that the sequence (EN)n∈Nof discretization grids is well distributed and well ordered (see Definition3.12), we denote byfNthe discretization of a homeomorphism f and byΩ(fN) the recurrent set offN(i.e.the union of periodic orbits offN).
We will show that for a lot of simple dynamical properties (P) about finite maps and for a generic conservative homeomorphism f, infinitely many discretizations fN satisfy (P) as well as infinitely many discretizations satisfy its contrary. For instance, for
4. That is, its entries are nonnegative and the sum of the elements of each of its columns and rows is equal to 1.
a generic homeomorphismf, the recurrent setΩ(fN) is sometimes as large as possible, i.e. Ω(fN) = EN (Corollary 5.9), sometimes very small (Corollary 5.15) and even bet-ter sometimes the number of elements of the image of EN is small (Corollary5.20). In the same way stabilization time5is sometimes zero (Corollary5.9, for example), some-times around Card(EN) (Corollary5.36). Finally, concerning the dynamics offN|Ω(fN), sometimes it is a cyclic permutation (Corollary5.9) or a bicyclic permutation (Corollary 5.13), sometimes it has many orbits (Corollary5.22). . .
Firstly, we deduce directly from Lax’s theorem that cyclic permutations of the sets ENform a dense type of approximation in Homeo(X,λ). Combining this with Theorem 5.2, we obtain directly:
Corollary 5.9(Miernowski, [Mie05]). For a generic homeomorphismf ∈Homeo(X,λ), for everyN0∈N, there existsN≥N0such thatfNis a cyclic permutation6.
This theorem states that for every generic conservative homeomorphism, there ex-ists a subsequence of discretizations which are “transitive”. Recall that generic homeo-morphism are transitive (see [Oxt37]). So, in a certain sense, transitivity can be detected on discretizations. Remark that this result implies that the discretizations of a generic conservative homeomorphism do not behave like typical random maps, as for a random map of a set withq elements the average number of periodic orbits is asymptotically logq(see for example [Bol01, XIV.5]).
There exists a variation of Lax’s Theorem for bicyclic permutations, which are per-mutations having exactly two orbits whose lengths are relatively prime (see [Gui12, lemme 2.9 page 28]); this variation leads to a proof of genericity in Homeo(X,λ) of topological weak mixing (see for example [AP00] or [Gui12, Part 2.4]). In the following we define a discrete analogue to topological weak mixing and state that this property occurs infinitely often on discretizations of a generic homeomorphism.
Definition 5.10. A homeomorphismf is said to betopologically weakly mixingif for all non-empty open sets (Ui)i≤M and (Ui0)i≤M, there exists m∈Nsuch thatfm(Ui)∩Ui0 is non-empty for alli≤M.
The proof of the genericity of topological weak mixing starts by an approximation of every conservative homeomorphism by another havingε-dense periodic orbits whose lengths are relatively prime. The end of the proof primarily involves the use of Baire’s theorem and Bézout’s identity. In the discrete case, the notion of weak mixing is re-placed by the following.
Definition 5.11. Letε>0. A finite mapσNis saidε-topologically weakly mixingif for all M∈Nand all balls (Bi)i≤Mand (B0i)i≤Mwith diameterε, there existsm∈Nsuch that for alli
σNm(Bi∩EN)∩(B0i∩EN),∅.
The first step of the proof is replaced by the following variation of Lax’s theorem:
5. That is, the smallest integerksuch thatfNk(EN) =Ω(fN).
6. In fact, T. Miernowski proves “permutation” but his arguments, combined with S. Alpern’s improve-ment of the Lax’s theorem, show “cyclic permutation”.
y0=y`× ×y1=f(y0) y2×
y3×
×y`−1 ×
fq1(y1)
× fq2(y2)
×
fq`−1(y`−1)
. . .
f
fq1 fq2
fq`−1
Figure 5.1: Construction of the sequence (ym)1≤m≤`
Proposition 5.12(First variation of Lax’s theorem). Letf ∈Homeo(X,λ)be a homeomor-phism such that all the iterates off are topologically transitive. Then for allε>0 and all M ∈ N∗, there exists N0 ∈ N such that for allN ≥ N0, there exists σN : EN →EN which has M ε-dense periodic orbits whose lengths are pairwise relatively prime, and such that dN(f ,σN)<ε.
Proof of Proposition5.12. We prove the proposition in the case where M = 2, the other cases being easily obtained by an induction. Letε>0 andf be a homeomorphism whose all iterates are topologically transitive. Then there existsx0 ∈X and p∈N∗ such that {x0,· · ·, fp−1(x0)} isε-dense andd(x0, fp(x0))<ε/2. Since transitive points offp form a dense Gδ subset of X, while the orbit ofx0form a Fσ set with empty interior, the set of points whose orbit underfp is dense and disjoint from that ofx0is dense. So we can pick such a transitive pointy0. Sety1=f(y0). Then there exists a multipleq1ofpsuch that the orbit {y1,· · ·, fq1−1(y1)} isε-dense and d(y1, fq1(y1))< ε/2. Again, by density, we can choose a transitive pointy2whose orbit is disjoint from that of x0andy1, with d(y1, y2)<ε/2 andd(y0, y1)−d(y0, y2)>ε/4. Then there exists a multipleq2ofpsuch that d(y2, fq2(y2))<ε/2. And so on, we construct a sequence (ym)1≤m≤` such that (see figure 5.1):
(i) for allm, there existsqm>0 such thatp|qmandd(ym, fqm(ym))<ε/2,
(ii) the orbits {x0,· · ·, fp−1(x0)} and {ym,· · ·, fqm−1(ym)} (m going from 0 to `−1) are pairwise disjoints,
(iii) for allm,d(ym, ym+1)<ε/2 andd(y0, ym)−d(y0, ym+1)>ε/4, (iv) y`=y0.
Let σN be a finite map given by Lax’s theorem. For all N large enough,σN satisfies the same properties (i) to (iii) than f. Changing σN at the points σNqm−1((ym)N) and σNp−1((x0)N), we obtain a finite mapσN0 such thatσN0 qN((ym)N) = (ym+1)NandσNp((x0)N) = (x0)N. Thus the orbit of (x0)N underσN0 is 2ε-dense and has periodpand the orbit of (y0)Nunderσ0Nis 2ε-dense, disjoint from which of (x0)Nand has period 1 +q1+· · ·+q`−1
relatively prime top.
Corollary 5.13. For a generic homeomorphismf ∈Homeo(X,λ), for allε>0and allN0∈N, there existsN≥N0such thatfNisε-topologically weakly mixing.
Proof of Corollary5.13. Again, we prove the corollary in the case where M = 2, other cases being easily obtained by induction. Letε>0 and N0∈N. All iterates of a generic homeomorphismf are topologically transitive: it is an easy consequence of the generic-ity of transitivgeneric-ity (see for example Corollary5.9or [Gui12, Theorem 2.11]); we pick such a homeomorphism. Combining Theorem 5.2and Proposition5.12, we obtain N≥N0 such thatfN has twoε/3-dense periodic orbits whose lengths petq are coprime. We now have to prove thatfN isε-topologically weakly mixing. Let B1, B2, B01and B02 be balls with diameter ε. Since each one of these orbits is ε/3-dense, there existsxN ∈X which is in the intersection of the orbit whose length isp and B1, andyN ∈ X which is in the intersection of the orbit whose length is qand B2. Similarly, there exists two integersaandbsuch thatfNa(xN)∈B01andfNb(yN)∈B02.
Recall that we want to find a power offNwhich sends both xN in B01 andyN in B02. It suffices to pickm∈Nsuch thatm=a+αp=b+βq. Bézout’s identity states that there exists two integersαandβsuchαp−βq=b−a. Setm=a+αp, adding a multiple ofpq if necessary, we can suppose thatmis positive. ThusfNm(xN)∈B01andfNm(yN)∈B02.
For now, the two approximation types we studied concern analogues of properties that are generic among Homeo(X,λ). We now show that some discrete analogues of properties that are not generic among conservative homeomorphisms also occur in-finitely often in the discretizations of generic homeomorphisms. The second variation of Lax’s theorem concerns the approximation of applications whose recurrent set is small.
Proposition 5.14(Second variation of Lax’s theorem). Letf ∈Homeo(X,λ). Then for all ε,ε0>0, there existsN0∈Nsuch that for allN≥N0, there exists a mapσN: EN→ENsuch thatdN(f ,σN)<εand
Card(Ω(σN))
Card(EN) =Card(Ω(σN)) qN <ε0, and such thatENis made of a unique (pre-periodic) orbit ofσN.
Proof of Proposition5.14. Letf ∈Homeo(X,λ),ε>0 and a recurrent pointxoff. There existsτ∈N∗such thatd(x, fτ(x))< 8ε; this inequality remains true for fine enough dis-cretizations: there exists N1∈Nsuch that if N≥N1, then
d(x, xN)< ε 8, d
fτ(x), fτ(xN)
< ε
8 and τ
qN <ε0.
Using the modulus of continuity offτand Lax’s theorem (Theorem 5.5), we obtain an integer N0≥ N1 such that for all N≥ N0, there exists a cyclic permutationσN of EN such thatdN(f ,σN)<2ε anddN(fτ,σNτ)<8ε. Then
d(xN,σNτ(xN))≤d(xN, x) +d
x, fτ(x) +d
fτ(x), fτ(xN) +d
fτ(xN),σNτ(xN)
<ε 2.
We composeσNby the (non bijective) application mappingσNτ−1(xN) onxNand being identity anywhere else (see Figure5.2), in other words we consider the application
σN0 (x) =
( xN ifx=στN−1(xN) σN(x) otherwise.
xN σN(xN)
σN2(xN)
σNτ−1(xN)
σNτ(xN)
στ+1N (xN)
σNqN−2(xN) σNqN−1(xN)
xN σN(xN)
σN2(xN)
σNτ−1(xN)
σNτ(xN)
στ+1N (xN)
σNqN−2(xN) σNqN−1(xN)
Figure 5.2: Modification of a cyclic permutation in the proof of Proposition5.14
The map σN0 has a unique injective orbit whose associated periodic orbit Ω(σN0 ) has length τ (it is (xN,σN(xN),· · ·,σNτ−1(xN))). Since d(f ,σN0 ) < ε, the map σN0 verifies the conclusions of the proposition.
A direct application of Theorem5.2leads to the following corollary.
Corollary 5.15. For a generic homeomorphismf ∈Homeo(X,λ), limN→+∞
Card(Ω(fN)) Card(EN) = 0.
Specifically for all ε > 0 and all N0 ∈ N, there exists N ≥ N0 such that Card(Ω(fN))/Card(EN) < ε and such that EN is made of a unique (pre-periodic) orbit of fN.
We now improve this corollary in stating that for a generic homeomorphism, there exists C >0 such that we have Card(Ω(fN))≤ C for an infinite number of orders N;
in particular these discretizations are highly non transitive, which is the opposite be-haviour to the dynamics of a generic homeomorphism.
The same kind of idea than in the proof of Proposition 5.14leads to the following variation of Lax’s theorem.
Proposition 5.16. Letf ∈ Homeo(X,λ) having at least one periodic point x of period p.
Then for allε>0, there exists N0∈Nsuch that for allN≥N0, there exists an application σN: EN →EN withdN(f ,σN)<ε, such that EN is made of a unique (pre-periodic) orbit of σNand such that the unique periodic orbit offN is of lengthpandε-shadows thef-orbit of x.
Proof of proposition5.16. Simply replace the recurrent point by a periodic point of pe-riodpin the proof of Proposition5.14.
We will use this proposition in Section5.5to obtain Theorem5.36. In particular, it will imply the following statement.
Corollary 5.17. For a generic homeomorphism f ∈ Homeo(X,λ), for every period p of a periodic point off and for infinitely many integersN,fN has a unique periodic orbit, whose length isp, and such thatENis covered by a single (pre-periodic) orbit offN.
Remark5.18. In particular, ifp0denotes the minimal period of the periodic points off, then there are infinitely many discretizations such that the cardinality of their recurrent set is equal top0. Note that however, the shortest period of periodic points of generic homeomorphisms has no global upper bound in Homeo(X,λ): for example, for allp∈N, there is an open set of homeomorphisms of the torus without periodic point of period less thanp(e.g.the neighbourhood of an irrational rotation) and this property remains true for discretizations.
Corollary5.17states that for a generic homeomorphism, Card(Ω(fN))/Card(EN) is as small as possible for an infinite number of orders N. The next result states that this loss of injectivity can even occur from the first iteration offN.
Proposition 5.19(Third variation of Lax’s theorem). Letf ∈Homeo(X,λ)andϑ:N→ R∗+a map which tends to+∞at+∞. Then for allε>0, there existsN0∈Nsuch that for all N≥N0, there exists a mapσN: EN→ENsuch thatCard(σN(EN))<ϑ(N)anddN(f ,σN)<ε.
Proof of Proposition5.19. Let f ∈Homeo(X,λ), ϑ : N→R∗+ a map which tends to +∞ at +∞andε>0. By Lax’s theorem (Theorem5.5) there exists N1∈Nsuch that for all N≥N1, there exists a cyclic permutationσN: EN→EN whose distance tof is smaller thanε/2. For N≥N1, setσ0N= PN1◦σN. Increasing N1if necessary we haved(f ,σN0 )<ε, regardless of N. Moreover Card(σN0 (EN))≤qN1; if we choose N0large enough such that for all N≥N0we haveqN1 <ϑ(N), then Card(σN0 (EN))≤ϑ(N). We have shown that the mapσN0 satisfies the conclusions of proposition for all N≥N0.
Corollary 5.20. Let ϑ : N → R∗+ a map which tends to +∞ at +∞. Then for a generic homeomorphismf ∈Homeo(X,λ),
lim
N→+∞
Card(fN(EN)) ϑ(N) = 0.
In particular, generically,limCard(fCard(EN(EN))
N) = 0.
Proof of Corollary5.20. Remark that if we replace ϑ(N) by p
ϑ(N), it suffices to prove that for a generic homeomorphism, limCard(fϑ(N)N(EN)) ≤1. This is easily obtained by com-bining Theorem5.2and Proposition5.19.
So far all variations of Lax’s theorem have constructed finite maps with a small num-ber of cycles. With the additional assumption that the sequence of grids is sometimes self similar, we show a final variation of Lax’s theorem, approaching every homeomor-phism by a finite map with a large number of orbits.
Proposition 5.21(Fourth variation of Lax’s theorem). Assume that the sequence of grids (EN)N∈Nis sometimes self similar. Let f ∈Homeo(X,λ) andϑ: N→Rsuch thatϑ(N) = o(qN). Then for all ε > 0, there exists N1 ∈ N such that for all N ≥ N1, there exists a permutationσNofENsuch thatdN(f ,σN)<εand that the number of cycles ofσNis greater thanϑ(N). Moreover,ϑ(N)of these cycles ofσNare conjugated to a cyclic permutation ofEN0 by bijections whose distance to identity is smaller thanε.
Proof of Proposition5.21. Let ε> 0, for all N0 ∈ N large enough, Lax’s theorem gives us a cyclic permutation σN0
0 of EN0 whose distance to f is smaller than ε. Since the grids are sometimes self similar, there exists N1 ∈Nsuch that for all N ≥N1, the set EN containsqN/qN0 ≥ϑ(N) disjoint subsetseEjN, each one conjugated to a grid EN0 by a bijectionhj whose distance to identity is smaller thanε. On eacheEjN, we defineσN as the conjugation ofσN0
0 byhj; outside these sets we just pickσNsuch thatdN(f ,σN)<ε.
Since the distance between hj and identity is smaller than ε, we have dN(f ,σN)< 2ε.
Moreover,σN has at leastϑ(N) cycles which are conjugated to a cyclic permutation of EN0by the mapshj; this completes the proof.
The application of Theorem5.2leads to the following corollary, which ensures that an infinite number of discretizationsfNhave a lot of periodic orbits.
Corollary 5.22. We still assume that the sequence of grids(EN)N∈Nis sometimes self similar.
Letϑ:N→Rsuch thatϑ(N) =o(qN). Then for a generic homeomorphismf ∈Homeo(X,λ) and for infinitely many integersN, the discretizationfNoff has at leastϑ(N)periodic orbits (which all have the same period).
If we compose the map obtained by Proposition5.21by an appropriate map of the finite set{hi(x0)}iinto itself (x0being fixed), we can prove the following result.
Proposition 5.23. Letf ∈Homeo(X,λ), ε>0and]a, b[⊂[0,1]. Then there exists N1∈N such that for allN≥N1, there exists a permutationσN ofEN such thatdN(f ,σN)<εand that the degree of recurrence ofσN(that is, the ratio between the cardinality of the recurrent set ofσNand the cardinality ofEN) belongs to]a, b[.
This proposition implies trivially the following corollary.
Corollary 5.24. For a generic conservative homeomorphismf ∈Homeo(X,λ), the sequence D(fN) of the degrees of recurrence of the discretizations accumulates on the whole segment [0,1].