Average behaviour of discretizations

We now want to study the average behaviour of discretizations of a generic homeo-morphism. For example, we could imagine that even if for a generic homeomorphism f, the event “f_N is a cyclic permutation” appears for infinitely many orders N, it is statistically quite rare.

More precisely, we study the frequency of occurrence of properties related to the discretizations of generic homeomorphisms: given a property (P) concerning discretiza-tions, we are interested in the behaviour of the proportion between 1 and M of dis-cretizations satisfying the property (P), when M goes to infinity. For this study, we assume that the sequence of discretization grids is always self similar (which is true for example for the torus equipped with discretizations upon uniform grids of orders powers of an integer, see Section3.7). This prevents us from having to deal with tricky arithmetic problems about overlay of grids.

Definition 5.25. Letf ∈Homeo(X,λ). We say that a property (P) concerning discretiza-tions issatisfied in average if for all N₀∈Nand allε>0, there exists N ≥N₀ such that

the proportion of integers M∈ {0,· · ·,N}such thatf_M satisfies (P) is greater than 1−ε,

We will show that most of the dynamical properties studied in the previous section are actually satisfied on average for generic homeomorphisms. To start with, we set out a technical lemma:

Lemma 5.26. LetT be a dense type of approximation inHomeo(X,λ). Then for a generic homeomorphismf ∈Homeo(X,λ), for allε>0and allα>0, the property(P): “E_Ncontains at least α disjoints subsets which fills a proportion greater than 1−εof E_N, each of them stabilized byf_N and such that the restriction off_Nto each one is conjugated to a map ofT by a bijection whose distance to identity is smaller thanε” is satisfied in average.

In practice, this lemma provides many properties satisfied on average, for instance:

– quantitative properties on discretizations, such as possessing at least M periodic orbits;

– properties of existence of sub-dynamics on discretizations, such as possessing at least one dense periodic orbit.

Proof of Lemma5.26. Let us consider the set C = \

It is simply obtained in combining the density of the type of approximationT and the fact that the sequence of grids is always self similar.

This lemma allows us to obtain some properties about the average behaviour of discretizations. For instance here is an improvement of Corollary5.36.

Corollary 5.27. For a generic homeomorphismf ∈Homeo(X,λ), the property “f_Nhas a ε-dense periodic orbit and the cardinality ofΩ(f_N) satisfiesCard(Ω(f_N)) =o(q_N)” is satisfied in average.

Or an improvement of Corollary5.13.

Corollary 5.28. For a generic homeomorphismf ∈Homeo(X,λ)and for allε>0, the prop-erty “f_N isε-topologically weakly mixing” (see Definition5.11) is satisfied in average.

Or even an improvement of Corollary5.20.

Corollary 5.29. For a generic homeomorphismf ∈Homeo(X,λ)and for allε>0, the prop-erty “^Card(f_Card(E^N(EN))

N) <ε” is satisfied in average.

And an improvement of Corollary5.22.

Corollary 5.30. For a generic homeomorphismf ∈Homeo(X,λ)and for allM∈N, property

“f_N has at leastMperiodic orbits” is satisfied in average.

However, note that the most simple property about discretizations,i.e.being a cyclic permutation, cannot be proved by using Lemma 5.26. To do this, we need a slightly more precise result, that requires the hypothesis that the grids are always strongly self similar.

Theorem 5.31. For a generic homeomorphismf ∈Homeo(X,λ), the property “f_Nis a cyclic permutation” is satisfied in average.

Note that this implies that most of the discretizations of f does not behave like a random map of a set of cardinalityq_N, as a random map of a set withqelements has in average logqperiodic orbits (see for example [Bol01, XIV.5]).

Lemma 5.32. Let f ∈ Homeo(X,λ) and ε > 0. The sequence of grids is supposed to be always strongly self similar. Then there existsN₀∈Nsuch that for allN≥N₀, there exists a permutationσ_NofE_N, such thatd_N(f ,σ_N)<ε,σ_N|E_N0 is a cyclic permutation ofE_N0 and for

Figure 5.3: Construction⁷ of Lemma 5.32 for two grids of orders 3 and 6 and zoom around a point of a grid of order 3

Proof of Lemma5.32. Applying Lax’s theorem (Theorem5.5), we can find an integer N₀ and a cyclic permutationσ_N

0 of E_N0 such thatd_N0(f ,σ_N

0)<ε. As the sequence of grids is always strongly self similar, we can suppose that N₀ is big enough to verify the con-clusions of Definition3.13.

Let us observe what happens for the order N₀+ 1. We will define an application σ_N⁰

0+1on E_N0+1, which will be close to the homeomorphismf. On E_N0, we defineσ_N⁰

0+1

as being equal toσ_N0. The idea is to repeat the proof of Lax’s theorem for the elements of E_N0+1\E_N0. To do that, we have to find a partition of X into sets with the same measure, such that every element A_x of this partition is one to one associated to a point

7. Do not forget that we identify some points of the boundary!

xof E_N0+1\E_N0, and has “small” diameter. So it suffices to split equitably the mass of the cubes E_N0 to the other cubes.

For this purpose, we cut each cube of E_N0+1corresponding to a pointx_N0of the grid E_N into α_N−1 subsets of the same measure 1/(q_N0+1(α_N−1)) (see Figure 5.3). Each of these subsets is assigned to the cube corresponding to the point h_i(x_N0). For each x∈E_N0+1\E_N0, we define A_x as the union of the cube C_x with the subset that is asso-ciated to it. Then all the sets A_x have the same measure and have a diameter smaller than twice the diameter of a cube of E_N0+1plusε. We can now apply Lax’s theorem to the homeomorphismf and the partition{A_x}_x∈E

N of X, this partition being numbered the same way as the partition E_N0+1 (some numbers are skipped); this gives a cyclic permutation of E_N0+1\E_N0 which is close tof. This cyclic permutation defines the per-mutationσ⁰_N

0+1where it was not yet. This σ_N⁰

0+1 is close tof and permutes cyclically the elements of E_N0and these of E_N0+1\E_N0.

We finish the proof by an induction, iterating N−N₀times.

Corollary 5.33. Let f ∈ Homeo(X,λ) and ε > 0 (we recall that the sequence of grids is supposed to be always strongly self similar). Then there exists N₀ ∈ N such that for all N ≥ N₀, there exists a homeomorphism g whose distance to f is smaller than ε such that for all M∈ {N,· · ·,N₀}, the discretizationg_M is a cyclic permutation ofE_M. Moreover, this property can be supposed to be verified on a neighbourhood ofg.

Proof of Corollary5.33. Letf ∈Homeo(X,λ) andε>0. Lemma5.32gives us an integer N₀∈Nsuch that for all N≥N₀, there exists a permutationσ_N of E_N, whose distance to f is smaller than ε/2, such that σ_N is a cyclic permutation of E_N0 and for all M ∈ {N₀+1,· · ·,N},σ_Mis a cyclic permutation of E_M\E_M−1. Set N≥N₀, the idea is to modify slightly the orbits ofσ_Nsuch thatσ_Nbecomes a cyclic permutation on all the sets E_M.

We begin by choosing two points x_N0 ∈ E_N0 and x⁰_N

0+1∈ E_N0+1such that x⁰_N

0+1 be-longs to the cube corresponding to the pointx_N0. Then, we modifyσ_Nin interchanging the pointsx_N0 etx⁰_N

0+1, in other words we set σ_N^N0(x) =











σ_N(x⁰_N

0+1) ifx=x_N0 σ_N(x_N0) ifx=x⁰_N

0+1

σ_N(x) otherwise.

Thus,σ_N^N0is a cyclic permutation of E_N0+1and the discretization of order N₀+1 ofσ_N^N0is also cyclic. We build the same way some mapsσ_N^N0+1,· · ·,σ_N^Nin interchanging the images of two adjacent points lying in the grids E_M et E_M+1. Then, for all M∈ {N₀,· · ·,N}, the discretization of σ_N^M−1on E_M is a cyclic permutation. Moreover, the distance between the map σ_N^N and σ_N is smaller than ε/2. Combined with Proposition 3.3, this allows us to interpolateσ_N^Nto a homeomorphismg whose distance tof is smaller thanεsuch that for all M ∈ {N₀,· · ·,N}, the discretization of order M of g is a cyclic permutation of E_M. A careful reader would notice that it may happen that in this construction, the discretizations ofg are not uniquely defined; depending on the choices made during the definition of P_Non E⁰_N, these discretizations may not be cyclic permutations. If we want to avoid this problem, it suffices to modify slightly the map σ_N^M; moreover this ensures that the conclusions of the corollary are verified on a whole neighbourhood of g.

Proof of Theorem5.31. Let the set homeo-morphismg⁰ close enough tog, we have

1

N + 1Cardn

M∈ {0,· · ·,N} |g_M⁰ ∈Po

>1−ε.

Such a homeomorphismgis simply given by Corollary5.33for N≥N₀such thatεk_N>

k_N0.

Remark5.34. However, the property of approximation by bicyclic permutations in av-erage cannot be proven directly with this technique.

Remark 5.35. A simple calculation shows that everything that has been done in this section also applies to the behaviour of discretizations in average of Cesàro average, in average of average of Cesàro average etc.,i.e.when studying quantities

1