Semi-hyperbolicity and hyperbolicity
Marcin Mazur
Jagiellonian University
Joint work with
J. Tabor and P. Kościelniak
Fifth Symposium on Nonlinear Analysis
Toruń 2007
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
1 / 10
Introduction
Motivation
Let M be a C 1 -manifold and f : M → M a diffeomorphism. We are given an
invariant subset K of M (i.e. f (K ) = K ).
Field of interest:
The behaviour of f in (the neighbourhood of) K .
A lot of information follows from the fact that K is hyperbolic for f .
Hyperbolicity:
guarantees stability, shadowing, expansivity, symbolic dynamics, etc.;
hard to verify.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
2 / 10
Introduction
Motivation
Let M be a C 1 -manifold and f : M → M a diffeomorphism. We are given an
invariant subset K of M (i.e. f (K ) = K ).
Field of interest:
The behaviour of f in (the neighbourhood of) K .
A lot of information follows from the fact that K is hyperbolic for f .
Hyperbolicity:
guarantees stability, shadowing, expansivity, symbolic dynamics, etc.;
hard to verify.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
2 / 10
Introduction
Motivation
Let M be a C 1 -manifold and f : M → M a diffeomorphism. We are given an
invariant subset K of M (i.e. f (K ) = K ).
Field of interest:
The behaviour of f in (the neighbourhood of) K .
A lot of information follows from the fact that K is hyperbolic for f .
Hyperbolicity:
guarantees stability, shadowing, expansivity, symbolic dynamics, etc.;
hard to verify.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
2 / 10
Introduction
Idea of hyperbolicity
By Tx M we denote the tangent space at the point x.
We consider a compact invariant set K ⊂ M and the collection of linear tangent
maps
Dx f : Tx M → Tf (x) M
for x ∈ K .
Roughly speaking, we say that K is hyperbolic if the linear operators Dorb(x) f
(x ∈ K ) are all hyperbolic ”with the same hyperbolicity constants”.
L∞ (orb(x)) 3 (vn )
..
..
..
.
.
.
Tf (x) M
3 v1
Tx M
3 v0
Tf −1 (x) M
3 v−1
..
..
..
.
.
.
7→
..
.
Dorb(x) f (vn )
..
.
7→
Df (x) f v1
7
→
Dx f v 0
7
→
Df −1 (x) f v−1
..
..
.
.
∈ L∞ (orb(x))
..
..
.
.
∈
∈
∈
..
.
Tf 2 (x) M
Tf (x) M
Tx M
..
.
We therefore go to the linear case −→ not so obvious!
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
3 / 10
Introduction
Idea of hyperbolicity
By Tx M we denote the tangent space at the point x.
We consider a compact invariant set K ⊂ M and the collection of linear tangent
maps
Dx f : Tx M → Tf (x) M
for x ∈ K .
Roughly speaking, we say that K is hyperbolic if the linear operators Dorb(x) f
(x ∈ K ) are all hyperbolic ”with the same hyperbolicity constants”.
L∞ (orb(x)) 3 (vn )
..
..
..
.
.
.
Tf (x) M
3 v1
Tx M
3 v0
Tf −1 (x) M
3 v−1
..
..
..
.
.
.
7→
..
.
Dorb(x) f (vn )
..
.
7→
Df (x) f v1
7
→
Dx f v 0
7
→
Df −1 (x) f v−1
..
..
.
.
∈ L∞ (orb(x))
..
..
.
.
∈
∈
∈
..
.
Tf 2 (x) M
Tf (x) M
Tx M
..
.
We therefore go to the linear case −→ not so obvious!
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
3 / 10
Introduction
Idea of hyperbolicity
By Tx M we denote the tangent space at the point x.
We consider a compact invariant set K ⊂ M and the collection of linear tangent
maps
Dx f : Tx M → Tf (x) M
for x ∈ K .
Roughly speaking, we say that K is hyperbolic if the linear operators Dorb(x) f
(x ∈ K ) are all hyperbolic ”with the same hyperbolicity constants”.
L∞ (orb(x)) 3 (vn )
..
..
..
.
.
.
Tf (x) M
3 v1
Tx M
3 v0
Tf −1 (x) M
3 v−1
..
..
..
.
.
.
7→
..
.
Dorb(x) f (vn )
..
.
7→
Df (x) f v1
7
→
Dx f v 0
7
→
Df −1 (x) f v−1
..
..
.
.
∈ L∞ (orb(x))
..
..
.
.
∈
∈
∈
..
.
Tf 2 (x) M
Tf (x) M
Tx M
..
.
We therefore go to the linear case −→ not so obvious!
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
3 / 10
Introduction
Idea of hyperbolicity
By Tx M we denote the tangent space at the point x.
We consider a compact invariant set K ⊂ M and the collection of linear tangent
maps
Dx f : Tx M → Tf (x) M
for x ∈ K .
Roughly speaking, we say that K is hyperbolic if the linear operators Dorb(x) f
(x ∈ K ) are all hyperbolic ”with the same hyperbolicity constants”.
L∞ (orb(x)) 3 (vn )
..
..
..
.
.
.
Tf (x) M
3 v1
Tx M
3 v0
Tf −1 (x) M
3 v−1
..
..
..
.
.
.
7→
..
.
Dorb(x) f (vn )
..
.
7→
Df (x) f v1
7
→
Dx f v 0
7
→
Df −1 (x) f v−1
..
..
.
.
∈ L∞ (orb(x))
..
..
.
.
∈
∈
∈
..
.
Tf 2 (x) M
Tf (x) M
Tx M
..
.
We therefore go to the linear case −→ not so obvious!
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
3 / 10
Introduction
Idea of hyperbolicity
By Tx M we denote the tangent space at the point x.
We consider a compact invariant set K ⊂ M and the collection of linear tangent
maps
Dx f : Tx M → Tf (x) M
for x ∈ K .
Roughly speaking, we say that K is hyperbolic if the linear operators Dorb(x) f
(x ∈ K ) are all hyperbolic ”with the same hyperbolicity constants”.
L∞ (orb(x)) 3 (vn )
..
..
..
.
.
.
Tf (x) M
3 v1
Tx M
3 v0
Tf −1 (x) M
3 v−1
..
..
..
.
.
.
7→
..
.
Dorb(x) f (vn )
..
.
7→
Df (x) f v1
7
→
Dx f v 0
7
→
Df −1 (x) f v−1
..
..
.
.
∈ L∞ (orb(x))
..
..
.
.
∈
∈
∈
..
.
Tf 2 (x) M
Tf (x) M
Tx M
..
.
We therefore go to the linear case −→ not so obvious!
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
3 / 10
Introduction
Definition of hyperbolicity – linear case
Let X be a Banach space and A : X → X a linear operator.
Definition
We say that A is hyperbolic if there exist a splitting X = Xs ⊕ Xu and an
equivalent norm k · k on X such that
Xs , Xu are A-invariant;
there exist λs < 1, λu > 1 such that
kAxs k ¬ λs kxs k, kAxu k ­ λu kxu k
for xs ∈ Xs , xu ∈ Xu .
Standard situation in R2 – saddle fixed point:
λs 0
A=
0 λu
λs < 1 < λ u
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
4 / 10
Introduction
Definition of hyperbolicity – linear case
Let X be a Banach space and A : X → X a linear operator.
Definition
We say that A is hyperbolic if there exist a splitting X = Xs ⊕ Xu and an
equivalent norm k · k on X such that
Xs , Xu are A-invariant;
there exist λs < 1, λu > 1 such that
kAxs k ¬ λs kxs k, kAxu k ­ λu kxu k
for xs ∈ Xs , xu ∈ Xu .
Standard situation in R2 – saddle fixed point:
λs 0
A=
0 λu
λs < 1 < λ u
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
4 / 10
Quasi-hyperbolicity
Hénon map
Problem:
Hyperbolicity of the Hénon attractor for various parameter values.
Partial Answer:
Theorem (Z. Arai)
Hénon attractor is hyperbolic on its chain recurrent part for a large set of
parameter values.
Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, to appear in Experimental Mathematics.
A computer assisted proof −→ the use of quasi-hyperbolicity.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
5 / 10
Quasi-hyperbolicity
Hénon map
Problem:
Hyperbolicity of the Hénon attractor for various parameter values.
Partial Answer:
Theorem (Z. Arai)
Hénon attractor is hyperbolic on its chain recurrent part for a large set of
parameter values.
Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, to appear in Experimental Mathematics.
A computer assisted proof −→ the use of quasi-hyperbolicity.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
5 / 10
Quasi-hyperbolicity
Hénon map
Problem:
Hyperbolicity of the Hénon attractor for various parameter values.
Partial Answer:
Theorem (Z. Arai)
Hénon attractor is hyperbolic on its chain recurrent part for a large set of
parameter values.
Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, to appear in Experimental Mathematics.
A computer assisted proof −→ the use of quasi-hyperbolicity.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
5 / 10
Quasi-hyperbolicity
Hénon map
Problem:
Hyperbolicity of the Hénon attractor for various parameter values.
Partial Answer:
Theorem (Z. Arai)
Hénon attractor is hyperbolic on its chain recurrent part for a large set of
parameter values.
Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, to appear in Experimental Mathematics.
A computer assisted proof −→ the use of quasi-hyperbolicity.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
5 / 10
Semi-hyperbolicity
How to deal with hyperbolicity?
Standard situation in R2 :
A=
λs
0
0
λu
λs < 1 < λ u
EU
ES
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
6 / 10
Semi-hyperbolicity
How to deal with hyperbolicity?
Standard situation in R2 :
A=
λs
?
?
λu
λs < 1 < λ u
EU
ES
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
6 / 10
Semi-hyperbolicity
How to deal with hyperbolicity?
Standard situation in R2 :
A=
λs
µu
µs
λu
λs < 1 < λu and (1 − λs )(λu − 1) > µs µu
EU
ES
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
6 / 10
Semi-hyperbolicity
How to deal with hyperbolicity?
One of possible solutions −→ semi-hyperbolicity:
easier to check;
allows lipschitzian perturbations.
A.A. Al-Nayef, P.E. Kloeden, A.V. Pokrovskii, Semi-hyperbolic mappings, condensing operators,
and neutral delay equations, J. Differential Equations 137 (1997), 320–339.
P. Diamond, P.E. Kloeden, V. S. Kozyakin, A.V. Pokrovskii, Semi-hyperbolic mappings, in
preparation.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
6 / 10
Semi-hyperbolicity
Definition of hyperbolicity – linear case (reformulation)
Let X be a Banach space, X = Xs ⊕ Xu .
We consider a linear operator A : X → X and assume that Xs and Xu are
A-invariant. Then in the matrix form we have
Ass
0
A=
.
0 Auu
Definition
We say that A is hyperbolic if
λs < 1 < λ u ;
1
kAs k ¬ λs , kA−1
u k ¬ λu ;
kAsu k ¬ µs , kAus k ¬ µu ;
for some equivalent norm k · k in X .
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
7 / 10
Semi-hyperbolicity
Definition of semi-hyperbolicity – linear case
Let X be a Banach space, X = Xs ⊕ Xu .
We consider a linear operator A : X → X and do not assume that Xs and Xu are
A-invariant. Then in the matrix form we have
Ass Asu
A=
.
Aus Auu
Definition
We say that A is semi-hyperbolic if
λs < 1 < λu and (1 − λs )(λu − 1) > µs µu ;
1
kAs k ¬ λs , kA−1
u k ¬ λu ;
kAsu k ¬ µs , kAus k ¬ µu ;
for some equivalent norm k · k in X .
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
7 / 10
Semi-hyperbolicity implies hyperbolicity
Results
Theorem
If the operator A is semi-hyperbolic, then A is hyperbolic.
M. Mazur, J. Tabor, K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case,
Proceedings of the Conference “Topological Methods in Differential Equations and Dynamical
Systems” (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math. 36 (1998), 121–126.
Theorem
Let K be a compact invariant semi-hyperbolic subset of M. Then K is hyperbolic.
M. Mazur, J. Tabor, P. Kościelniak, Semi-hyperbolicity and hyperbolicity, to appear in Discrete
Contin. Dynam. Syst. (IM UJ preprint 2007/04, http://www.im.uj.edu.pl/badania/preprinty/).
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
8 / 10
Semi-hyperbolicity implies hyperbolicity
Results
Theorem
If the operator A is semi-hyperbolic, then A is hyperbolic.
M. Mazur, J. Tabor, K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case,
Proceedings of the Conference “Topological Methods in Differential Equations and Dynamical
Systems” (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math. 36 (1998), 121–126.
Theorem
Let K be a compact invariant semi-hyperbolic subset of M. Then K is hyperbolic.
M. Mazur, J. Tabor, P. Kościelniak, Semi-hyperbolicity and hyperbolicity, to appear in Discrete
Contin. Dynam. Syst. (IM UJ preprint 2007/04, http://www.im.uj.edu.pl/badania/preprinty/).
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
8 / 10
Semi-hyperbolicity implies hyperbolicity
Results
Theorem
If the operator A is semi-hyperbolic, then A is hyperbolic.
M. Mazur, J. Tabor, K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case,
Proceedings of the Conference “Topological Methods in Differential Equations and Dynamical
Systems” (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math. 36 (1998), 121–126.
Theorem
Let K be a compact invariant semi-hyperbolic subset of M. Then K is hyperbolic.
M. Mazur, J. Tabor, P. Kościelniak, Semi-hyperbolicity and hyperbolicity, to appear in Discrete
Contin. Dynam. Syst. (IM UJ preprint 2007/04, http://www.im.uj.edu.pl/badania/preprinty/).
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
8 / 10
Semi-hyperbolicity implies hyperbolicity
Results
Theorem
If the operator A is semi-hyperbolic, then A is hyperbolic.
M. Mazur, J. Tabor, K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case,
Proceedings of the Conference “Topological Methods in Differential Equations and Dynamical
Systems” (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math. 36 (1998), 121–126.
Theorem
Let K be a compact invariant semi-hyperbolic subset of M. Then K is hyperbolic.
M. Mazur, J. Tabor, P. Kościelniak, Semi-hyperbolicity and hyperbolicity, to appear in Discrete
Contin. Dynam. Syst. (IM UJ preprint 2007/04, http://www.im.uj.edu.pl/badania/preprinty/).
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
8 / 10
Semi-hyperbolicity implies hyperbolicity
Main results
Theorem (MM, J. Tabor, P. Kościelniak)
Let M be a Riemannian manifold, f : M → M be a C 1 -diffeomorphism and K be
an invariant subset of M. Assume that K is (λs , λu , µs , µu ; h)-semi-hyperbolic. Let
γs∗ , γu∗ be arbitrary reals such that
√
√
(λu −λs )2 −4µs µu
(λu −λs )2 −4µs µu
λs +λu
λs +λu
∗
∗
−
<
1
<
γ
<
<
γ
+
.
s
u
2
2
2
2
Let
(λu −λs +µs +µu )h
u −λs +µs +µu )h
C ∗ = max{ (γ ∗(λ
−λs )(λu −γ ∗ )−µs µu , (γ ∗ −λs )(λu −γ ∗ )−µs µu }.
s
Then the set K is
s
u
u
(γs∗ , γu∗ ; C ∗ )-hyperbolic.
Theorem (MM, J. Tabor)
Hénon attractor is hyperbolic for some parameter values.
A computer assisted proof −→ the C++ program that verifies the
semi-hyperbolicity conditions for the Hénon attractor.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
9 / 10
Semi-hyperbolicity implies hyperbolicity
Main results
Theorem (MM, J. Tabor, P. Kościelniak)
Let M be a Riemannian manifold, f : M → M be a C 1 -diffeomorphism and K be
an invariant subset of M. Assume that K is (λs , λu , µs , µu ; h)-semi-hyperbolic. Let
γs∗ , γu∗ be arbitrary reals such that
√
√
(λu −λs )2 −4µs µu
(λu −λs )2 −4µs µu
λs +λu
λs +λu
∗
∗
−
<
1
<
γ
<
<
γ
+
.
s
u
2
2
2
2
Let
(λu −λs +µs +µu )h
u −λs +µs +µu )h
C ∗ = max{ (γ ∗(λ
−λs )(λu −γ ∗ )−µs µu , (γ ∗ −λs )(λu −γ ∗ )−µs µu }.
s
Then the set K is
s
u
u
(γs∗ , γu∗ ; C ∗ )-hyperbolic.
Theorem (MM, J. Tabor)
Hénon attractor is hyperbolic for some parameter values.
A computer assisted proof −→ the C++ program that verifies the
semi-hyperbolicity conditions for the Hénon attractor.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
9 / 10
Semi-hyperbolicity implies hyperbolicity
Main results
Theorem (MM, J. Tabor, P. Kościelniak)
Let M be a Riemannian manifold, f : M → M be a C 1 -diffeomorphism and K be
an invariant subset of M. Assume that K is (λs , λu , µs , µu ; h)-semi-hyperbolic. Let
γs∗ , γu∗ be arbitrary reals such that
√
√
(λu −λs )2 −4µs µu
(λu −λs )2 −4µs µu
λs +λu
λs +λu
∗
∗
−
<
1
<
γ
<
<
γ
+
.
s
u
2
2
2
2
Let
(λu −λs +µs +µu )h
u −λs +µs +µu )h
C ∗ = max{ (γ ∗(λ
−λs )(λu −γ ∗ )−µs µu , (γ ∗ −λs )(λu −γ ∗ )−µs µu }.
s
Then the set K is
s
u
u
(γs∗ , γu∗ ; C ∗ )-hyperbolic.
Theorem (MM, J. Tabor)
Hénon attractor is hyperbolic for some parameter values.
A computer assisted proof −→ the C++ program that verifies the
semi-hyperbolicity conditions for the Hénon attractor.
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
9 / 10
Final remarks
Work for (near) future
improvement of the HENON-HYPER program: stricter estimations, interval
parameters, etc.;
development and implementation of the algorithm for systems in R3 ;
looking for interesting examples.
http://www.im.uj.edu.pl/MarcinMazur/comphyp/
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
10 / 10
Final remarks
Work for (near) future
improvement of the HENON-HYPER program: stricter estimations, interval
parameters, etc.;
development and implementation of the algorithm for systems in R3 ;
looking for interesting examples.
http://www.im.uj.edu.pl/MarcinMazur/comphyp/
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
10 / 10
Final remarks
Work for (near) future
improvement of the HENON-HYPER program: stricter estimations, interval
parameters, etc.;
development and implementation of the algorithm for systems in R3 ;
looking for interesting examples.
http://www.im.uj.edu.pl/MarcinMazur/comphyp/
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
10 / 10
Final remarks
Work for (near) future
improvement of the HENON-HYPER program: stricter estimations, interval
parameters, etc.;
development and implementation of the algorithm for systems in R3 ;
looking for interesting examples.
http://www.im.uj.edu.pl/MarcinMazur/comphyp/
Marcin Mazur (Jagiellonian University)
Semi-hyperbolicity and hyperbolicity
SNA 2007
10 / 10