[Paper Review] A (short) survey on Dominated Splitting

Martı́n Sambarino|arXiv (Cornell University)|Mar 24, 2014

Mathematical Dynamics and Fractals54 references20 citations

TL;DR

This survey provides a comprehensive overview of dominated splitting in smooth dynamical systems, emphasizing its role in robust dynamical behavior and structural stability. It establishes that dominated splitting is preserved under small C¹ perturbations and demonstrates the existence of non-hyperbolic systems with dominated splitting through a recursive C¹-perturbation construction, yielding a C¹-robust non-Anosov diffeomorphism on the 2-torus with periodic points whose stable Lyapunov exponents tend to zero.

ABSTRACT

We present here the concept of Dominated Splitting and give an account of some important results on its dynamics.

Motivation & Objective

• To systematize the theory of dominated splitting as a foundational concept in smooth dynamics.
• To clarify the role of dominated splitting in ensuring robust dynamical phenomena under C¹ perturbations.
• To investigate the limitations of dominated splitting in implying hyperbolicity or structural stability.
• To construct explicit examples of C¹ diffeomorphisms with dominated splitting that are not Anosov, using iterative perturbations.
• To explore the implications of dominated splitting for generic C¹ dynamics and homoclinic classes.

Proposed method

• Utilizes a recursive C¹-perturbation technique to modify Anosov diffeomorphisms on the 2-torus while preserving dominated splitting.
• Applies local coordinate changes and cone field arguments to control the norm of the derivative on invariant subbundles.
• Employs Lyapunov exponent estimates and cone conditions to ensure domination: ‖Df|E‖ < e^γ and ‖Df|F‖ < e^γ with γ < 0.
• Constructs a sequence of diffeomorphisms {f_n} converging in C¹ topology to a limit g with dominated splitting.
• Uses the fact that periodic points outside the perturbation support inherit hyperbolicity from earlier f_n, ensuring g is expansive.
• Applies the theory of expansive homeomorphisms to conclude that g is conjugate to an Anosov diffeomorphism despite not being Anosov.

Experimental results

Research questions

• RQ1 Can dominated splitting be preserved under small C¹ perturbations, and what dynamical consequences does it imply?
• RQ2 To what extent does dominated splitting imply structural stability or hyperbolicity in dynamical systems?
• RQ3 Can one construct a C¹ diffeomorphism with dominated splitting that is not Anosov, and if so, what are its dynamical properties?
• RQ4 How does the presence of dominated splitting affect the existence and index of periodic points in homoclinic classes?
• RQ5 What role does smoothness play in determining whether a dominated splitting leads to hyperbolicity or robust dynamical behavior?

Key findings

• A C¹-robust non-Anosov diffeomorphism g on the 2-torus exists with dominated splitting, constructed as the C¹-limit of Anosov diffeomorphisms.
• The limit diffeomorphism g has a sequence of periodic points p_n with stable Lyapunov exponents L^s(p_n, g) = -λ_n → 0 as n → ∞.
• Despite not being Anosov, g is expansive and thus conjugate to an Anosov diffeomorphism, showing that dominated splitting does not imply hyperbolicity.
• The stable foliation tangent to E^s is preserved throughout the perturbation process, ensuring the persistence of the dominated splitting.
• The construction shows that homoclinic classes with dominated splitting E^s ⊕ E^c ⊕ E^u can exist without periodic points of index dim E^s, even after perturbation.
• The result illustrates that dominated splitting alone is insufficient to guarantee hyperbolicity or the existence of periodic attractors, even under C¹ generic conditions.