Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 1/22

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1 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 1/22 Super-exponential growth of the number of periodic orbits inside homoclinic classes Todd Fisher Department of Mathematics University of Maryland, College Park

2 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 2/22 Abstract Joint work with Christian Bonatti and Lorenzo Diaz.

3 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 2/22 Abstract Joint work with Christian Bonatti and Lorenzo Diaz. We will show that C 1 generically if a homoclinic class contains periodic points of different indices, then super-exponential growth of periodic points inside the homoclinic class.

4 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 3/22 Artin-Mazur diffeomorphisms Definition: A diffeomorphism f is Artin-Mazur (A-M) if the number of isolated periodic points of period n of f, denoted Per(n, f) grows exponentially fast.

5 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 3/22 Artin-Mazur diffeomorphisms Definition: A diffeomorphism f is Artin-Mazur (A-M) if the number of isolated periodic points of period n of f, denoted Per(n, f) grows exponentially fast. There exists C > 0 such that #Per(n, f) exp(cn) for all n N.

6 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 3/22 Artin-Mazur diffeomorphisms Definition: A diffeomorphism f is Artin-Mazur (A-M) if the number of isolated periodic points of period n of f, denoted Per(n, f) grows exponentially fast. There exists C > 0 such that #Per(n, f) exp(cn) for all n N. Artin-Mazur( 65) proved A-M maps are dense in the space of diffeomorphisms.

7 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 4/22 Genericity Definition: A set R X is generic (residual) if it contains a set that is the countable intersection of open and dense sets in X.

8 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 4/22 Genericity Definition: A set R X is generic (residual) if it contains a set that is the countable intersection of open and dense sets in X. For diffeomorphisms generic sets are dense. Corresponds to topologically typical.

9 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 5/22 Kaloshin s result Kaloshin( 00) showed A-M not generic for r 2. Follows by looking at Newhouse domain (unfolding of tangency).

10 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 5/22 Kaloshin s result Kaloshin( 00) showed A-M not generic for r 2. Follows by looking at Newhouse domain (unfolding of tangency). Technical point in Kaloshin s proof says an open set K of diffeomorphisms has super-exponential growth of periodic points generically if for every sequence of positive integers a = {a n } n=1 there is a generic subset R(a) of K such that lim sup n #Per(n, f)/a n = for any f R(a).

11 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 6/22 Homoclinic Class Definition: For a hyperbolic periodic point p of f the homoclinic class H(p) = W s (Op) W u (Op).

12 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 6/22 Homoclinic Class Definition: For a hyperbolic periodic point p of f the homoclinic class H(p) = W s (Op) W u (Op). Defined by Newhouse( 72) as generalization of hyperbolic basic sets.

13 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 6/22 Homoclinic Class Definition: For a hyperbolic periodic point p of f the homoclinic class H(p) = W s (Op) W u (Op). Defined by Newhouse( 72) as generalization of hyperbolic basic sets. We will show generically the superexponential growth occurs inside a homoclinic class.

14 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 7/22 Linked periodic points Consider a C 1 open set of diffeomorphism U such that each f U contains hyperbolic periodic saddles p f and q f depending continuously on f.

15 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 7/22 Linked periodic points Consider a C 1 open set of diffeomorphism U such that each f U contains hyperbolic periodic saddles p f and q f depending continuously on f. From (ABCDW, preprint) there is a generic set G of U such that either either H(p f, f) = H(q f, f) for all f G, or H(p f, f) H(q f, f) = for all f G.

16 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 7/22 Linked periodic points Consider a C 1 open set of diffeomorphism U such that each f U contains hyperbolic periodic saddles p f and q f depending continuously on f. From (ABCDW, preprint) there is a generic set G of U such that either either H(p f, f) = H(q f, f) for all f G, or H(p f, f) H(q f, f) = for all f G. First case called generically homoclinically linked g.h.l.

17 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 8/22 Generically linked Note: Homoclinically related for periodic points says W s (p) W u (q) and W u (p) W s (q). This is an open condition, but is different than generically homoclinically linked.

18 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 8/22 Generically linked Note: Homoclinically related for periodic points says W s (p) W u (q) and W u (p) W s (q). This is an open condition, but is different than generically homoclinically linked. Definition: The index of a hyperbolic periodic point is the dimension of E s (p).

19 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 8/22 Generically linked Note: Homoclinically related for periodic points says W s (p) W u (q) and W u (p) W s (q). This is an open condition, but is different than generically homoclinically linked. Definition: The index of a hyperbolic periodic point is the dimension of E s (p). Remark: Homoclinically related periodic points have the same index, but generically homoclinincally linked peridoic points can have different indices.

20 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 9/22 Theorems Theorem 1(Bonatti, Diaz, F.) There is a residual subset S(M) of Diff 1 (M) of diffeomorphisms f such that, for every f S(M) any homoclinic class of f containing hyperbolic periodic saddles of different indices has super-exponential growth of the number of periodic points.

21 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 9/22 Theorems Theorem 1(Bonatti, Diaz, F.) There is a residual subset S(M) of Diff 1 (M) of diffeomorphisms f such that, for every f S(M) any homoclinic class of f containing hyperbolic periodic saddles of different indices has super-exponential growth of the number of periodic points. We can actually show if indices are α and β with α < β then for each γ [α, β] there is super-exponential growth of periodic points of index γ in the homoclinic class.

22 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 10/22 Corollary Corollary: Every non-hyperbolic homoclinic class of a C 1 -generic diffeomorphism with a finite number of homoclinic classes has super-exponential growth of the number of periodic points.

23 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 10/22 Corollary Corollary: Every non-hyperbolic homoclinic class of a C 1 -generic diffeomorphism with a finite number of homoclinic classes has super-exponential growth of the number of periodic points. Remark: In (ABCDW) it is shown that C 1 generically the indices of saddles in the homoclinic class form an interval in N.

24 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 11/22 Hyperbolic homoclinic class For a hyperbolic homoclinic class Λ we know h top (f Λ ) = lim sup n log #P (n, f) n. Then the periodic points have exponential growth equal to topological entropy.

25 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 11/22 Hyperbolic homoclinic class For a hyperbolic homoclinic class Λ we know h top (f Λ ) = lim sup n log #P (n, f) n. Then the periodic points have exponential growth equal to topological entropy. In fact if Λ is topologically mixing c 1, c 2 > 0 such that for all n. c 1 e nh top(f Λ ) #P (n, f) c 2 e nh top(f Λ )

26 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 12/22 Star Diffeomorphisms Definition: A diffeomorphism f is a star diffeomorphism if it has a neighborhood U in Diff 1 (M) such that each g U has only hyperbolic periodic points.

27 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 12/22 Star Diffeomorphisms Definition: A diffeomorphism f is a star diffeomorphism if it has a neighborhood U in Diff 1 (M) such that each g U has only hyperbolic periodic points. For C 1 diffeomorphisms this is equivalent to Axiom A plus no-cycles (Hayashi( 97) and Aoki( 92)).

28 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 12/22 Star Diffeomorphisms Definition: A diffeomorphism f is a star diffeomorphism if it has a neighborhood U in Diff 1 (M) such that each g U has only hyperbolic periodic points. For C 1 diffeomorphisms this is equivalent to Axiom A plus no-cycles (Hayashi( 97) and Aoki( 92)). C 1 generic diffeomorphisms in complement of star diffeomorphisms have super-exponential growth, but not necessarily in homoclinic class.

29 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 13/22 Prevalence There are examples (easy to construct) of generic sets with zero Lebesgue measure. Measure theoretically typical and topologically typical don t always agree. Definition: For a one parameter family of diffeomorphisms f t a property is prevalent if it holds on a set of full Lebesgue measure.

30 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 13/22 Prevalence There are examples (easy to construct) of generic sets with zero Lebesgue measure. Measure theoretically typical and topologically typical don t always agree. Definition: For a one parameter family of diffeomorphisms f t a property is prevalent if it holds on a set of full Lebesgue measure. Problem:(Arnold) For a (Baire) generic finite parameter family of diffeomorphisms f t, for Lebesgue almost every t we have that f t is A-M.

31 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 14/22 Idea from ABCDW If there are g.h.l. points p and q of index α and α + 1 respectively, then after a perturbation there is a saddle-node r with dim(e s (r)) = n α 1, dim(e u (r)) = α, dim(e c (r)) = 1.

32 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 14/22 Idea from ABCDW If there are g.h.l. points p and q of index α and α + 1 respectively, then after a perturbation there is a saddle-node r with dim(e s (r)) = n α 1, dim(e u (r)) = α, dim(e c (r)) = 1. Furthermore, W s (r) W u (q) and W u (r) W s (p).

33 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 15/22 Picture of p,q, and r q p s

34 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 16/22 Idea of proof Note: In ABCDW r can be chosen of arbitrarily large period and E c (r) near identity.

35 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 16/22 Idea of proof Note: In ABCDW r can be chosen of arbitrarily large period and E c (r) near identity. Let a = {a n } n=1 be a given sequence. Let r be of period n. After perturbation r is identity in center direction so we can perturb to get na n saddles of index α (and na n of index α + 1)

36 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 17/22 Creation of periodic points s s

37 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 18/22 Idea of proof - part 2 For each k N there exists a residual set G α (k) such that each g G(k) there exists n g (k) k where H(p g, g) has n g (k)a ng (k) different periodic points of index α.

38 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 18/22 Idea of proof - part 2 For each k N there exists a residual set G α (k) such that each g G(k) there exists n g (k) k where H(p g, g) has n g (k)a ng (k) different periodic points of index α. Now let R α (a) = k Gα (k). This is residual.

39 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 19/22 Important Claim Claim: For each f R α (a) it holds that lim sup k #Per α (f, k) a k =.

40 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 19/22 Important Claim Claim: For each f R α (a) it holds that lim sup k #Per α (f, k) a k =. Idea of proof. Since f G α (k) for all k N n f (k) k such that H(p, f) contains n f (k)a nf (k) points of index α. So there is a subsequence n k such that #(Per α (f) H(p f, f)) a nk n k.

41 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 20/22 End of proof Now let R(a) = β α=γ Rγ (a). The set R(a) is residual in U and growth of saddles is lower bounded by a.

42 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 21/22 Conjecture Residually homoclinic classses depend continuously on diffeomorphism and number of homoclinic classes is locally constant. A diffeomorphism is wild if the number of homoclinic classes is locally infinite.

43 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 21/22 Conjecture Residually homoclinic classses depend continuously on diffeomorphism and number of homoclinic classes is locally constant. A diffeomorphism is wild if the number of homoclinic classes is locally infinite. Conjecture: There is a C 1 generic dichotomy for diffeomorphisms: either the homoclinic classes are hyperbolic or there is a super-exponential growth of the number of periodic points.

44 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 21/22 Conjecture Residually homoclinic classses depend continuously on diffeomorphism and number of homoclinic classes is locally constant. A diffeomorphism is wild if the number of homoclinic classes is locally infinite. Conjecture: There is a C 1 generic dichotomy for diffeomorphisms: either the homoclinic classes are hyperbolic or there is a super-exponential growth of the number of periodic points. Finite number done by corollary

45 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 22/22 Symbolic Extensions Definition: A diffeomorphism has a symbolic extension if there exists shift space (Σ, σ) and factor map π : Σ M such that fπ = πσ.

46 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 22/22 Symbolic Extensions Definition: A diffeomorphism has a symbolic extension if there exists shift space (Σ, σ) and factor map π : Σ M such that fπ = πσ. Question: Among diffeomorphisms containing a homoclinic class with periodic points of differing index is there a C 1 -residual set S such that for any f S does not have a symbolic extension?

47 Super-exponential growth of the number of periodic orbits inside homoclinic classes p. 22/22 Symbolic Extensions Definition: A diffeomorphism has a symbolic extension if there exists shift space (Σ, σ) and factor map π : Σ M such that fπ = πσ. Question: Among diffeomorphisms containing a homoclinic class with periodic points of differing index is there a C 1 -residual set S such that for any f S does not have a symbolic extension? Preprint and copy of this presentation at tfisher