Super-exponential growth
of the number of periodic
orbits inside homoclinic
classes
Todd Fisher
[email protected]
Department of Mathematics
University of Maryland, College Park
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 1/22
Abstract
Joint work with Christian Bonatti and Lorenzo
Diaz.
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.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 2/22
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.
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.
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.
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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.
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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.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 4/22
Kaloshin’s result
Kaloshin(’00) showed A-M not generic for r ≥ 2.
Follows by looking at Newhouse domain
(unfolding of tangency).
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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 = {an }∞
n=1 there
is a generic subset R(a) of K such that
lim supn→∞ #Per(n, f )/an = ∞ for any f ∈ R(a).
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 5/22
Homoclinic Class
Definition: For a hyperbolic periodic point p of f
the homoclinic class H(p) = W s (Op) t W u (Op).
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) t W u (Op).
Defined by Newhouse(’72) as generalization of
hyperbolic basic sets.
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) t 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.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 6/22
Linked periodic
points
Consider a C 1 open set of diffeomorphism U
such that each f ∈ U contains hyperbolic
periodic saddles pf and qf depending
continuously on f .
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 pf and qf depending
continuously on f .
From (ABCDW, preprint) there is a generic set G
of U such that either
either H(pf , f ) = H(qf , f ) for all f ∈ G,
or H(pf , f ) ∩ H(qf , f ) = ∅ for all f ∈ G.
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 pf and qf depending
continuously on f .
From (ABCDW, preprint) there is a generic set G
of U such that either
either H(pf , f ) = H(qf , f ) for all f ∈ G,
or H(pf , f ) ∩ H(qf , f ) = ∅ for all f ∈ G.
First case called generically homoclinically linked
g.h.l.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 7/22
Generically linked
Note: Homoclinically related for periodic points
says W s (p) t W u (q) and W u (p) t W s (q). This is
an open condition, but is different than
generically homoclinically linked.
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) t W u (q) and W u (p) t 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 Es (p).
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) t W u (q) and W u (p) t 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 Es (p).
Remark: Homoclinically related periodic points
have the same index, but generically
homoclinincally linked peridoic points can have
different indices.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 8/22
Theorems
Theorem 1(Bonatti, Diaz, F.) There is a residual
1
subset S(M ) of Diff (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.
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
1
subset S(M ) of Diff (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.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 9/22
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.
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.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 10/22
Hyperbolic
homoclinic class
For a hyperbolic homoclinic class Λ we know
log #P (n, f )
htop (f |Λ ) = lim sup
.
n
n→∞
Then the periodic points have exponential growth
equal to topological entropy.
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Hyperbolic
homoclinic class
For a hyperbolic homoclinic class Λ we know
log #P (n, f )
htop (f |Λ ) = lim sup
.
n
n→∞
Then the periodic points have exponential growth
equal to topological entropy.
In fact if Λ is topologically mixing ∃ c1 , c2 > 0 such
that
c1 enhtop (f |Λ ) ≤ #P (n, f ) ≤ c2 enhtop (f |Λ )
for all n.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 11/22
Star
Diffeomorphisms
Definition: A diffeomorphism f is a star
diffeomorphism if it has a neighborhood U in
Diff1 (M ) such that each g ∈ U has only hyperbolic
periodic points.
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Star
Diffeomorphisms
Definition: A diffeomorphism f is a star
diffeomorphism if it has a neighborhood U in
Diff1 (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)).
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
Diff1 (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.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 12/22
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 ft a property is prevalent if it
holds on a set of full Lebesgue measure.
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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 ft 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 ft , for
Lebesgue almost every t we have that ft is A-M.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 13/22
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(Es (r)) = n − α − 1,
dim(Eu (r)) = α, dim(Ec (r)) = 1.
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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(Es (r)) = n − α − 1,
dim(Eu (r)) = α, dim(Ec (r)) = 1.
Furthermore, W s (r) t W u (q) 6= ∅ and
W u (r) t W s (p) 6= ∅.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 14/22
Picture of p,q, and r
q
p
s
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 15/22
Idea of proof
Note: In ABCDW r can be chosen of arbitrarily
large period and Ec (r) near identity.
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 Ec (r) near identity.
Let a = {an }∞
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 nan saddles of
index α (and nan of index α + 1)
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 16/22
Creation of periodic
points
s
s
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 17/22
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 ng (k) ≥ k
where H(pg , g) has ng (k)ang (k) different periodic
points of index α.
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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 ng (k) ≥ k
where H(pg , g) has ng (k)ang (k) different periodic
points of index α.
T α
α
Now let R (a) = k G (k). This is residual.
Super-exponential growth of the number of periodic orbits inside homoclinic classes – p. 18/22
Important Claim
Claim: For each f ∈ Rα (a) it holds that
α
#Per (f, k)
lim sup
= ∞.
ak
k→∞
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Important Claim
Claim: For each f ∈ Rα (a) it holds that
α
#Per (f, k)
lim sup
= ∞.
ak
k→∞
Idea of proof. Since f ∈ G α (k) for all k ∈ N ∃
nf (k) ≥ k such that H(p, f ) contains nf (k)anf (k)
points of index α. So there is a subsequence n k
such that
#(Perα (f ) ∩ H(pf , f ))
≥ nk .
a nk
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End of proof
Tβ
Now let R(a) = α=γ Rγ (a). The set R(a) is
residual in U and growth of saddles is lower
bounded by a.
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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.
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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.
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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
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Symbolic Extensions
Definition: A diffeomorphism has a symbolic
extension if there exists shift space (Σ, σ) and
factor map π : Σ → M such that f π = πσ.
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Symbolic Extensions
Definition: A diffeomorphism has a symbolic
extension if there exists shift space (Σ, σ) and
factor map π : Σ → M such that f π = 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?
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Symbolic Extensions
Definition: A diffeomorphism has a symbolic
extension if there exists shift space (Σ, σ) and
factor map π : Σ → M such that f π = 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
http://www.math.umd.edu/˜tfisher
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