Chaos and quasi-periodicity in diffeomorphisms of the solid torus

3.2.1 Perturbations of multimodal families
Two definitions from [14, Section 5.2] are introduced now. For more information about the
terminology, we refer to [24, Sections II.5, II.6].
Definition 4. Let J ⊂ be a compact interval. Fix d ≥ 1, k ≥ 3, a ∗ ∈ , and an interval
of parameter values U = [a − , a + ], with a ∗ ∈ Int U. A C k -family of maps M a : J → J, with
a ∈ U, is called a d-family if it satisfies the following conditions:
1. Invariance: M a ∗(J) ⊂ Int(J);
2. Nondegenerate critical points: M a ∗ has d critical points {c 1 , . . . , c d } def
= Cr M a ∗ that satisfy
M ′′
a ∗(c i) ≠ 0 for all i and M a ∗(c i ) ≠ c j for all i, j;
3. Negative Schwarzian derivative: SM a ∗ < 0 for all x ≠ c i , where
Sf(x) = f ′′′ (x)
f ′ (x) − 3 ( ) f ′′ 2
(x)
;
2 f ′ (x)
4. Topological mixing: There exists an interval J ′ ⊂ Int(J) such that M a ∗(J) = M a ∗(J ′ ) = J ′
and such that for any open intervals J 1 , J 2 in J ′ there exists n 0 such that
M n a ∗(J 1) ∩ J 2 ≠ ∅ for all n ≥ n 0 ;
5. Preperiodicity: for each 1 ≤ i ≤ d there exists m i such that p i = M m i
a ∗ (c i) is a (repelling)
periodic point of M a ∗;
6. Genericity of unfolding: For all c i ∈ Cr M a ∗, denote by c i (a) and p i (a) the continuations
of c i and p i , respectively, for a close to a ∗ . Then
d (
M
m i
a
da
(c i(a)) − p i (a) ) ≠ 0 at a = a ∗ .
Next we introduce the notion of η-perturbation of a d-family M a , with a ∈ U and d ≥ 1 fixed.
Definition 5. Fix σ > 0 and consider the family M a obtained by extending M a as follows:
M a : J × I σ → J × I σ ,
M a (x, y) def
=(M a (x), 0). (21)
Also denote by M the map
M : U × J × I σ → J × I σ ,
M(a, x, y) def
= M a (x, y) = (M a (x), 0).
Given a C k -family of diffeomorphisms
for a k ≥ 3, denote by G its extension
G a : J × I σ → J × I σ , a ∈ J,
G : U × J × I σ → J × I σ ,
G(a, x, y) def
= G a (x, y).
Then G is called a η-perturbation of the d-family {M a } a if
‖M − G‖ C k ≤ η,
where ‖·‖ C k denotes the C k -norm over U × J × I σ .
20