Ergodic Properties of C Generic Diffeomorphisms

Ergodic Properties of C1 Generic Diffeomorphisms Student: Bruno Santiago Advisor: Alexander Arbieto IM-UFRJ Approximation of Invariant Measures by Periodic Orbits Bonatti, Gourmelon and Vivier, [2], have a perturbation result wich allow us to alter the derivative of the diffeomorphism along a periodic orbit in order to make it have all eigenvalues equal in some The shadowing property, given by the ergodic closing lemma of Man˜e´ [3] is very usefull and when direction of the tangent space, where lack of domination appears. Using this tool we can prove the combined with some other hypotesis can be very powerfull. For example, let us analise the case following when f has an ergodic measure µ. 0.3 Corollary. Let f be a genric diffeomorphism and µ an ergodic measure for f . Define Λ := In this case, by Birkhoff Ergodic Theorem, for a generic point x Supp(µ) the foward orbit is dense Supp(µ) and let 2 in the support of µ. Once we shadow this orbit by a periodic orbit, as long as the shadowing is T M = F ::: F Λ 1 ⊕ ⊕ k sufficiently fine, we get that this periodic orbit is Hausdorff close of Supp(µ). This alredy gives be the finest dominated splitting. Then, there exists a sequence of periodic orbits xn Per( f ), such a slightly improvment in the ergodic closing lemma. A natural question is wheter we can also that 2 approximate the Lyapunov Exponents of µ by those of this periodic orbit. The answer is yes, but we have to work a little more in order to establish this. • µx µ in the weak topology; n ! 0.1 Proposition. Lef µ be an ergodic measure of a diffeomorphism f . Take a neighborhood of • O(x ) Λ in the Hausdorff topology; U n f , a neighborhood V of µ in the space (M) of invariant measures endowed with the weak-star ! R f log det D f F dµ M • for each i k, the Lyapunov exponents of inside F converge to j j ij . topology, a neighborhood U of Λ := Supp(µ) in the space of compact subsets of M endowed = 1; :::; µxn i λFi = F CM dim i with the Hausdorff topolgy, and a neighborhood O of the Lyapunov vector L(µ) of µ in Rd. Then The results we have so far deal only with ergodic measures. For increase their worth, we must there exist g having a periodic point x such that the periodic measure µx belongs to V, the 2 U ensure that set of ergodic measure is big enough for the study of the whole space of invariant orbit O(x) belongs to U and the Lyapunov vector of µ belongs to O. x measures. In the case of an Axiom A diffeomorphism this is true, at least in the topological sense, Lk by a result of Sigmund [4]. In our generic case we have the same conclusion for measures sup- Proof. Take an > 0 small, and take a generic point x Supp(µ). Suppose that TxM = E is 2 l=1 l ported on compact, isolated and transitive invariant sets. Since every periodic measure is ergodic, the Oseledts splitting of µ at x, and let n > 0 have the following properties: 0 it suffices to work with them. 1. for every l = 1; :::; k, v E implies 1 log D f n(x)v λ < , for every n n . 0.4 Theorem. Let Λ be an isoleted non-trivial transitive set of a C1-generic diffeomorphism f . 2 l n − l ≥ 0 n Then the set (Λ) of periodic measures supported on Λ is dense in the set (Λ) of invariant 2. the string x; :::; f (x) is -dense in the support of µ P f M f f g measures supported on Λ. 3. if V = V(φ ; :::; φr; δ) and g is such that x Per(g) with period n n and the g-orbit of x 1 2 U 2 ≥ 0 -shadows the f -orbit of x until the n-th iterate then Generic Measures for Generic Diffeomorphisms n 1 Z Now we shall look for properties of generic invariant measures for a generic diffeomorphism. Since 1 X− j φl(g (x)) φldµ < δ; the results we have so far fits better when the invarant measure is also ergodic, is natural try to n − j=0 show that ergodicity is a generic property in the space of inavariant measures. It turns out that the set of ergodic measures of a continuous transformation on a compact space is always a Gδ for every l = 1; :::; r: erg 0.5 Proposition. Let f Diff1(M) and Λ M a compact invariant. Then, the set (Λ) of 2 ⊂ M f The two first items follows, resp., from Osledts and Birkhoff theorems. The third one is obteined ergodic invariant measures supported on Λ is a G subset of (Λ). δ M f combining continuity of the functions φl, using the shadowing, and Birkhoff Ergodic Theorem. In virtue of theorem 0.4, if f and Λ, in the statement of theorem 0.4, are resp. generic and isoleted Since x is gereric, by the ergodic closing lemma there exist g , such that x is a periodic point erg 2 U then (Λ) turns into a residual subset of f (Λ). of g with period n n0 and the g-orbit of x -shadows the f -orbit of x until the n-th iterate. By items M f M ≥ (2) and (3) above, we have that the periodic measure µx belongs to V and the orbit O(x) belongs We also wish to look for measures that have a big support. This is true for generic measures, as to U. We only have to perturb g a little bit to control the Lyapunov exponents. the next result shows. By the shadowing property we can use local charts arround the f -orbit of x, and compare Dg(gj(x)) 0.6 Proposition. Let f and Λ be as in the proposition 0.5. Then, for every generic measure µ in with D f ( f j(x)). By shrinking if necessary, we may assume that these maps are close. Using (Λ) we have U M f [ isometries A : T j M T j M, we have that the maps j g (x) ! f (x) Supp(µ) = Supp(ν): ν f (Λ) 1 j 2M Lj = A−j+1D f ( f (x))Aj Again, if f and Λ are resp. generic and isoleted then we have that generic invariant measures are close to Dg(gj(x)). Franks Lemma gives us a diffeomorphism g , that prserves the g-orbit are fully supported in Λ. Using these results and the fact that for generic ergodic measures the j 2 U of x, and achieve the map Lj as derivative at the point g (x). Moreover the exponets of the peridoic Lyapunov exponents vary continuously, we can prove the following result measure of g are given by 0.7 Theorem. Let f and Λ be as in theorem 0.4. Then, every generic µ f (Λ) is ergodic, fully supported, the Osledets Splitting is dominated and all Lyapunov exponents2 M are non-zero. 1 1 n log σ = log D f (x)v (λi , λi + ); n j j n 2 − Proof. We fix arbitraly a subbundle F of the finest dominated splitting over Λ. where σ is an eigenvalue of Dgn(x), and λ is one of the Lyapunov exponents of µ. This shows that By the results above we have a residual subset of (Λ) such that every measure in is i R M f R the Lyapunov vector of µx is -close to the Lyapunov vector of µ, and completes the proof. ergodic, fully supported and is a continuity point of the map L : ν L(ν). We shall add one more property to this residual: the sums of the Lyapunov exponents of7! each of the measures inside F Once we have this perturbative result, we can make it generic and in this case we don’t need to are non-zero. perturb the diffeomorphism to obtain the required periodic measures. The argument (in spirit) is the same that genarates the General Density Theorem of Phug from the closing lemma. For this prurpose, first note that by Franks Lemma, given any diffeomorphism f and any periodic point p of f , we can perturb f along the orbit of p to make all the sums λ + ::: + λ , of Lyapunov 0.2 Theorem. Given an ergodic measure µ of a C1-generic diffeomorphism f , there is a sequence i j exponents of µp, 0 i j d, non-zero. Hence, for every integer n, the set xn Per( f ) such that ≤ ≤ ≤ 2 n 1 o • the measures µ converge to µ in the weak-star topology; An = f KS ; p Per( f ); π(p) n; I(µp) , 0 ; xn 2 8 2 ≤ • the periodic orbits O(xn) converge to Supp(µ) in the Hausdorff topology; R 1 where I(µp) = log det D f F dµp, is dense in Diff (M). Moreover, by the continuation of hyperbolic • the Lyapunov vectors L(µ ) converge to the Lyapunov vector L(µ). j j j xn periodic orbits, and since the function log det D(:) is continuous, we have that for every g suffi- j jFj 1 ciently close to f , all the continuations pg of periodic points p Per( f ), with π(p) n also satisfy Proof. Take a contable basis n for the topology of Diff (M). In each of these open sets we shall 2 ≤ I(µ ) 0.

Load more