26
3 Grids, discretizations, measures
3.1 The manifold X and the measureλ . . . 36 3.2 Generic properties. . . 36 3.3 C0extension of finite maps. . . 37 3.4 Discretization grids, discretizations of a homeomorphism . . . 37 3.5 Probability measures on X . . . 38 3.6 Hypothesis on discretization grids. . . 40 3.7 Some examples of discretization grids . . . 41 4 Discretizations of a generic dissipative homeomorphism
4.1 Dynamics of a generic dissipative homeomorphism . . . 45 4.2 Dynamics of discretizations of a generic homeomorphism . . . 49 4.3 Numerical simulations . . . 53 4.3.1 Algorithm used for the calculus of invariant measures . . . 54 4.3.2 Combinatorial behaviour. . . 55 4.3.3 Behaviour of invariant measures . . . 56 5 Discretizations of a generic conservative homeomorphism
5.1 Dense types of approximation . . . 62 5.2 Lax’s theorem . . . 64 5.3 Individual behaviour of discretizations . . . 65 5.4 Average behaviour of discretizations . . . 71 5.5 Behaviour of all the discretizations . . . 75 5.5.1 Periodic orbits . . . 76 5.5.2 Invariant measures . . . 78 5.5.3 Invariant compact sets . . . 83 5.6 Physical dynamics . . . 84 5.7 Addendum: generic conjugates and generic grids . . . 86 5.8 Numerical simulations . . . 87 5.8.1 Combinatorial behaviour. . . 88 5.8.2 Behaviour of invariant measures . . . 90 5.8.3 Periodic points. . . 96
6 How roundofferrors help to compute the rotation set
6.1 Notations and preliminaries . . . 101 6.1.1 Rotation sets . . . 101 6.1.2 Observable measures . . . 102 6.2 Observable rotation sets . . . 103 6.2.1 Definitions . . . 103 6.2.2 Properties of the observable rotation set . . . 105 6.3 Discretized rotation sets . . . 107 6.4 Numerical simulations . . . 109
In this part, we consider the dynamics of discretizations of generic homeomor-phisms,i.e.we tackle the following question:
Question. Which dynamical properties of a generic homeomorphism f can be read on the dynamics of its discretizations(fN)N≥0?
We will establish properties for both dissipative,i.e. arbitrary homeomorphisms of X, andconservativehomeomorphisms,i.e. homeomorphisms of X that preserve a given good probability measure. In this part, our results concern generic homeomorphisms of a compact manifold (with boundary) of dimensionn≥2.
We will prove many results, concerning various aspects of the dynamics of the dis-cretizations, adapting some classical tools of study of the generic dynamics of home-omorphisms. Morally, our results express that in the dissipative generic case, the dy-namics of the discretizations tends to the “physical” dydy-namics of the initial homeomor-phism4whereas in the conservative generic setting, the dynamics of the discretizations accumulates on all the possible dynamics of the initial homeomorphism, and moreover the physical dynamics cannot be detected on discretizations. In the rest of this intro-duction, we try to organize our results according to some “lessons”:
1) The dynamics of discretizations of a generic dissipative homeomorphism tends to the “physical dynamics” of the initial homeomorphism.
We first study properties of discretizations of genericdissipativehomeomorphisms5. The ergodic behaviour of such a generic homeomorphism is deduced from the shred-ding lemmaof F. Abdenur and M. Andersson [AA13] (Lemma4.2), which implies that a generic homeomorphism has a “attractor dynamics” (see Corollaries4.5and4.8):
Theorem A. For a generic homeomorphismf, the closureA0of the set of Lyapunov-stable periodic points is a Cantor set of dimension 0 which attracts almost every point ofX. More-over, the measureµfX(see Definition3.10page39) is well defined, atomless and is supported byA0.
This behaviour easily transmits to discretizations, for example every attractor of the homeomorphism can be seen on all the fine enough discretizations (Proposition4.13).
4. That is, the dynamics that occur for almost every point with respect to Lebesgue measure.
5. Without assumption of preservation of a given measure.
29
Theorem B. For a generic homeomorphismf, the recurrent set of its discretizationΩ(fN) tends to A0 in the following weak sense: for all ε > 0, there exists N0 ∈ N such 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,Ω(fe N))<ε.
Moreover, the Cesàro limit of the pushforwards of the uniform measures on the grid by the discretizations tend to the Cesàro limit of the pushforwards of λ by the homeomorphism. Indeed, we will prove the following (Theorem4.16).
Theorem C. For a generic homeomorphismf, the measures m1Pm−1
i=0 f∗iλconverge to a mea-sure that we denote byµfX (see Definition3.10).
Concerning the discretizations, for a generic homeomorphismf, the measuresµfXN tend to the measureµfX whenNgoes to infinity, whereµfXN is the measure on the periodic orbits offN such that the global measure of each periodic orbit is proportional to the size of its basin of attraction (see Definition3.10).
Moreover, there is shadowingof the dynamics off by that of its discretizationsfN (Corollary4.12).
Theorem D. For a generic homeomorphism f, for all ε > 0 and all δ >0, there is a full measure dense open subsetOofXsuch that for allx∈O, allδ>0and allNlarge enough, the orbit ofxN underfNδ-shadows6the orbit ofxunderf.
Thus, it is possible to detect on discretizations the “physical” dynamics of a generic dissipative homeomorphism, that is the dynamics that can be seen by almost every point of X. This dynamics is mainly characterized by the position of the attractors and of the corresponding basins of attraction.
2) The dynamics of a single discretization of a generic conservative homeomorphism cannot be inferred from the dynamics of the initial homeomorphism.
We then turn to the study of the conservative case. The starting point of our study is a question from É. Ghys (see [Ghy94, Section 6]): for a generic conservative homeo-morphism of the torus, what is the asymptotical behaviour of the sequence of degrees of recurrence offN? A partial answer to this question was obtained by T. Miernowski in the second chapter of his thesis (see Corollary5.9).
Theorem E (Miernowski). For a generic conservative homeomorphism7 f, there are in-finitely many integersNsuch that the discretizationfNis a cyclic permutation.
To prove this theorem, T. Miernowski combines a genericity argument with a quite classical technique in generic dynamics: Lax’s theorem [Lax71] (Theorem5.5), which states that any conservative homeomorphism can be approximated by cyclic permuta-tions of the discretization grids. In fact this proof can be generalized to obtain many results about discretizations. We will establish some variants of Lax theorem; each of them, combined with a genericity argument, leads to a result for discretizations of generic homeomorphisms. For instance, we will prove the following (Corollary5.36).
6. That is, for allk∈N,d(fNk(xN), fk(x))<δ.
7. That is, there is a Gδdense subset of the set of conservative homeomorphisms of the torus on which the conclusion of the theorem holds.
Theorem F. For a generic conservative homeomorphismf, there exists infinitely many inte-gersNsuch that the cardinality ofΩ(fN)is equal to the smallest period of periodic points of f.
Note that the combination of these two theorems answer É. Ghys’s question: for a generic homeomorphismf, the sequence of the degrees of recurrence offNaccumulates on both 0 and 1; we can even show that it accumulates on the whole segment [0,1]
(Corollary5.24).
Another variation of Lax’s theorem leads to a theorem that throws light on the be-haviour of the discretizations on their recurrent set (Corollary5.22).
Theorem G. For a generic conservative homeomorphism f and for all M ∈ N, there are infinitely many integersN such thatfN is a permutation of EN having at least Mperiodic orbits.
To summarize, generically, infinitely many discretizations are cyclic permutations, but also infinitely many discretizations are highly non-injective or else permutations with many cycles. In particular, it implies that for allx∈X, there exists infinitely many integers N such that the orbit ofxN underfN does not shadow the orbit ofx underf: in this sense, generically, the dynamics of discretizations does not reflect that of the homeomorphism. Note that this behaviour is in the opposite of the dissipative case, here the individual behaviour of discretizations does not indicate anything about the actual dynamics of the homeomorphism.
3) A dynamical property of a generic conservative homeomorphism cannot be deduced from the frequency it appears on discretizations either.
The previous theorems express that the dynamics of a single discretization does not reflect the actual dynamics of the homeomorphism. However, we might reasonably ex-pect that the properties of the homeomorphism are transmitted to many discretizations.
More precisely, we may hope that given a property (P) about discretizations, if there are many integers N such that the discretizationfN satisfies (P), then the homeomorphism satisfies a similar property. It is not so, for instance, we will prove the following result (Theorem5.31).
Theorem H. For a generic conservative homeomorphism f, when M goes to infinity, the proportion of integersNbetween1andMsuch thatfN is a cyclic permutation accumulates on both0and1.
In fact, for all the properties considered in the previous paragraph, the frequency with which they appear on discretizations of orders smaller than M accumulates on both 0 and 1 when M goes to infinity. Remark that these result imply that, even by looking at the frequency at which some properties occur, the discretizations of a generic conservative homeomorphism do not behave like typical random maps, as for a random map of a set withqelements, the average number of periodic orbits is asymptotically logq(see for example [Bol01, XIV.5]).
4) Many dynamical properties of a generic conservative homeomorphism can be de-tected by looking at the dynamics ofall the discretizations.
We have observed that we cannot detect the dynamics of a generic homeomorphism when looking at the dynamics of its discretizations, or even at the frequency with which
some dynamics appears on discretizations. Nevertheless, the dynamical properties of a generic conservative homeomorphism can be deduced from the analogous dynamical properties ofall the discretizations. More precisely, we have a shadowing property of the dynamical properties of the homeomorphism: for each dynamical property of the homeomorphism, its discrete analogue can be seen on an infinite number of discretiza-tions. It is worthwhile to note the intriguing fact that this shadowing property occurs for all the dynamical properties of a generic conservative homeomorphism, indepen-dently of the measureλ, while for a generic dissipative homeomorphism the dynam-ics of the discretizations converges to the “physical” dynamdynam-ics of the homeomorphism (that is, the dynamics depending ofλ).
This idea of convergence of the dynamics when looking at arbitrary large precisions can be related to the work of P. Diamondet al(see page20). For instance, we will prove that the periodic orbits of a homeomorphism can be detected by looking at the periodic orbits of its discretizations (Theorem5.36).
Theorem I. For a generic homeomorphismf, for everyε>0and every periodic orbit of f, this periodic orbit isε-shadowed by an infinite number of periodic orbits of the same period of the discretizations
We will also prove a theorem in the same vein for invariant measures (respectively invariant compact sets), which expresses that the set of invariant measures (respectively compact sets) of the homeomorphism can be deduced from the sets of invariant mea-sures (respectively invariant sets) of its discretizations. More precisely, we will prove the following result (Theorems5.45and5.51, see Theorems5.49and5.55for the com-pact versions).
Theorem J. For a generic conservative homeomorphismf and for every convex closed set (for Hausdorfftopology)M off-invariant Borel probability measures there exists an increasing sequence of integers Nk such that the set offNk-invariant probability measures tends toM (for Hausdorfftopology).
Moreover, ifM is reduced to a single measure, thenfNk can be supposed to bear a unique invariant measure.
In the third chapter of this part, we will present an application of the notion of discretization to the practical problem of computing numerically the rotation set of a torus homeomorphism. In particular, we will prove a theorem which expresses the shadowing property of the rotation set of a generic conservative torus homeomorphism by the rotation sets of its discretizations (Corollary6.23).
Theorem K. Iff is a generic conservative homeomorphism, then there exists a subsequence fNi of discretizations such thatρNi(F) tends toρ(F)for Hausdorfftopology (whereFis a lift off toR2).
This will give us a convenient way to compute the rotation set in practice, as we will prove that if we compute the rotation set corresponding to a starting pointx∈T2 without roundofferror, then for almost everyx∈T2, the obtained rotation set is reduced to a single vector, that is the mean rotation vector off (Proposition6.19).
Theorem L. Letf be a generic conservative homeomorphism of the torusT2andFa lift of f toR2. Then for almost everyx˜∈R2, the corresponding rotation set
ρ(x) = \
M∈N
[
m≥M
(Fm( ˜x)−x˜ m
)
is reduced to the mean rotation vector with respect to the measureλ.
This shows that if we try to compute the rotation set by calculating segments of or-bitswithoutmaking any roundoff error, we will only find the mean rotation vector of the homeomorphism. In Chapter6, we will introduce the notion of observable rotation set, which expresses which rotation vectors can be found by looking at almost every pe-riodic orbit. We will compute this set for some examples, in particular for both generic conservative (that is, TheoremL) and dissipative (Proposition6.15) homeomorphisms.
5) The “physical dynamics” of a generic conservative homeomorphism plays no partic-ular role for discretizations.
The heuristic idea underlying the concept of physical measure is that these mea-sures are the invariant ones which can be detected “experimentally” (since many initial conditions lead to these measures). Indeed, some experimental results on specific ex-amples of dynamical systems show that they are actually the measures that are detected in practice for these examples (see for example [BCG+78,BCG+79] or [Boy86,GB88]).
Moreover, if the dynamical system is uniquely ergodic, then the invariant measure ap-pears naturally on discretizations (see [Mie06, Proposition 8.1] and Proposition5.42).
According to this heuristic and these results, we could expect from physical mea-sures to be the only invariant meamea-sures that can be detected on discretizations of generic conservative homeomorphisms. This is not the case: for a generic conservative homeo-morphism, there exists a unique physical measure, namelyλ(that follows directly from the celebrated Oxtoby-Ulam’s theorem [OU41]). According to the previous theorem, invariant measures of the discretizations accumulate on all the invariant measures of the homeomorphism and not only on Lebesgue measure.
However, we could still hope to distinguish the physical measure from other invari-ant measures. For this purpose, we define the canonical physical measureµfXNassociated to a discretizationfN: it is the limit in the sense of Cesàro of the images of the uniform measure on ENby the iterates offN: ifλNis the uniform measure on EN, then
µfXN= lim
m→∞
1 m
m−1
X
i=0
(fNi)∗λN.
This measure is supported by the recurrent set Ω(fN); it is uniform on every periodic orbit and the total weight of a periodic orbit is proportional to the size of its basin of attraction. The following theorem expresses that these measures accumulate on the whole set of f-invariant measures: physical measures cannot be distinguished from other invariant measures on discretizations, at least for generic homeomorphisms (see Theorems5.51and5.53).
Theorem M. For a generic conservative homeomorphism f, the set of limit points of the sequence(µfXN)N∈Nis the set of allf-invariant measures. Also, for everyf-invariant measure µ, there exists a subsequencefNk of discretizations such that for everyx∈X, the sequence of measures8µfxNk tends toµ.
8. Recall that by Definition3.10,µfxN is the Cesàro limit of the pushforwards of the Dirac measureδxN by the discretizationfN.
The same phenomenon appears for compact invariant subsets: the the recurrent subsets of the discretizationsfNaccumulate on the whole set of invariant compact sub-sets forf (Proposition5.55).
6) On the numerical experiments we performed, the dynamics of a dissipative home-omorphism can be detected on discretizations, and a lot of different dynamical be-haviours can be observed on discretizations of a conservative homeomorphism.
We will compare our theoretical results with the reality of numerical simulations.
Indeed, it is not clear that the behaviour predicted by our results can be observed on computable discretizations of a homeomorphism defined by a simple formula. On the one hand, all our results are valid “forgenerichomeomorphisms”; nothing indicates that these results apply to actual examples of homeomorphisms defined by simple formulas.
On the other hand, results such as “there are infinitely many integers N such that the discretization of order N. . . ” provide no control over the integers N involved; they may be so large that the associated discretizations are not computable in practice.
We first carried out simulations of dissipative homeomorphisms. The results of dis-cretizations of a small perturbation of identity (in C0topology) may seem disappointing at first sight: the trapping regions of the initial homeomorphisms cannot be detected, and there is little difference with the conservative case. This behaviour is similar to that highlighted by J.-M. Gambaudo and C. Tresser in [GT83] (see page 2.2). That is why it seemed to us useful to test a homeomorphism which is C0close to the identity, but whose basins are large enough. In this case the simulations point to a behaviour that is very similar to that described by theoretical results, namely that the dynamics converges to the dynamics of the initial homeomorphism. In fact, we have actually observed behaviours as described by theorems only for examples of homeomorphisms with a very small number of attractors.
For conservative homeomorphisms, our numerical simulations produce mixed re-sults. From a quantitative viewpoint, the behaviour predicted by our theoretical result cannot be observed on our numerical simulations. For example, we do not observe any discretization whose degree of recurrence is equal to 1 (i.e. which is a permutation).
This is nothing but surprising: the events pointed out by the theorems area priorivery rare. For instance, there is a very little proportion of bijective maps among maps from a given finite set into itself. From a more qualitative viewpoint, the behaviour of the simulations is quite in accordance with the predictions of the theoretical results. For example, for a given conservative homeomorphism, the degree of recurrence of a dis-cretization depends a lot on the size of the grid used for the disdis-cretization. Similarly, the canonical invariant measure associated with a discretization of a homeomorphism f does depend a lot on the size of the grid used for the discretization.
7) Discretizations can actually be very useful and efficient to compute some dynamical invariants like the rotation set of a torus homeomorphism.
We have also performed numerical simulations of rotation sets. To obtain numeri-cally an approximation of the observable rotation set, we have calculated rotation vec-tors of long segments of orbits for a lot of starting points, these points being chosen randomly fore some simulations and being all the points of a grid on the torus for other simulations. For the numerical approximation of the asymptotic discretized rotation set we chosed a fine enough grid on the torus and have calculated the rotation vectors of periodic orbits of the discretization of the homeomorphism on this grid.
We made these simulations on an example where the rotation set is known to be the square [0,1]2. It makes us sure of the shape of the rotation set we should obtain numerically, however it limits a bit the “genericity” of the examples we can produce.
In the dissipative case we made attractive the periodic points which realize the ver-tex of the rotation set [0,1]2. It is obvious that these rotation vectors, which are realized by attractive periodic points with basin of attraction of reasonable size, will be detected by the simulations of both observable and asymptotic discretized rotation sets; that is we observe in practice: we can recover quickly the rotation set in both cases.
In the conservative setting we observe the surprising behaviour predicted by the theory: when we compute the rotation vectors of long segments of orbits we obtain mainly rotation vectors which are quite close to the mean rotation vector, in particular we do not recover the initial rotation set. More precisely, when we perform simulations with less than one hour of calculation we only obtain rotation vectors close to the mean rotation vector, and when we let three hours to the computer we only recover one vertex of the rotation set [0,1]2. On the other hand, the rotation set is detected very quickly by the convex hulls of discretized rotation sets (less than one second of calculation).
Moreover, when we calculate the union of the discretized rotation sets over several grids to obtain a simulation of the asymptotic discretized rotation set, we obtain a set which
Moreover, when we calculate the union of the discretized rotation sets over several grids to obtain a simulation of the asymptotic discretized rotation set, we obtain a set which