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On relationships between the Lyapunov spectrum and the Morse spectrum

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We consider dynamical systems on manifolds and explore the relationship between the Lyapunov and the Morse spectra. It is known that the Morse spectrum contains the Lyapunov spectrum and the Morse spectra are closed intervals whose boundary points are Lyapunov exponents. We investigate about the equality of the two spectra. In a compact invariant set in R2 we consider chain recurrent components of a continuous time flow. For planar flows there are only three possible types of nonwandering sets: fixed point, periodic orbit and cycle. We consider the situation when the nonwandering set coincides with the chain recurrent set and select these three kinds as isolated chain recurrent components. Over these three types of sets we perform a case by case study by first linearizing the system over the solutions and then applying the necessary techniques. The Lyapunov spectrum over a fixed point consists of the real parts of the eigenvalues of the Jacobian matrix of partial derivatives. For periodic orbit, they are the characteristic exponents. For a cycle we find that the Lyapunov spectrum consists of the Lyapunov exponents for the fixed points in the cycle. Selgrade's theorem gives a relationship between the chain recurrent components in the base and the chain recurrent components for the flow in the projective bundle. We use this theorem to find Morse spectrum. It turns out that for an isolated fixed point, a connected set of fixed points and a periodic orbit the two spectra coincide. For a cycle, when all the fixed points are hyperbolic, or when all of them have two distinct Lyapunov exponents, we get an attractor-repeller pair which gives two chain recurrent components. These correspond to two Morse intervals. For a fixed point in a cycle if both the Lyapunov exponents are the same then they can only take the value zero. When for at least one fixed point in a cycle both the Lyapunov exponents are zero the entire projective bundle turns out to be a single chain recurrent component. This gives a single Morse interval whose boundaries consist of the minimum and the maximum of the Lyapunov exponents. Hence we get intervals for Morse spectrum here whereas Lyapunov spectrum were just points. We conclude that the Morse spectrum is actually different from the Lyapunov spectrum.