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Exponential growth of separation on a bounded attractor is compatible with boundedness because the relevant measurement is local transverse stretching with renormalization along the attractor, not unlimited separation in the ambient space.

Conditions: The trajectory lies on an attractor; Separation is measured transversely with renormalization

Exponential growth of separation on a bounded attractor is compatible with boundedness because the relevant measurement is local transverse stretching with renormalization along the attractor, not unlimited separation in the ambient space.

Conditions: The trajectory lies on an attractor; Separation is measured transversely with renormalization

For general trajectories not confined exactly to a single stable or unstable manifold, the lecture argues that the largest positive Lyapunov exponent controls the long-time behavior. This is summarized on the slide as $\lambda_1$ dominating as $t\to\infty$.

Conditions: Trajectory experiences a mix of stable and unstable effects.; Looking at the long-time limit.

For general trajectories not confined exactly to a single stable or unstable manifold, the lecture argues that the largest positive Lyapunov exponent controls the long-time behavior. This is summarized on the slide as $\lambda_1$ dominating as $t\to\infty$.

Conditions: Trajectory experiences a mix of stable and unstable effects.; Looking at the long-time limit.