An introduction to the theory of smooth dynamical systems
Discusses the theoretical aspects and results of smooth dynamical systems. Covers dynamical systems on manifolds of one or two dimensions, generic properties, stability theory, invariant measures for differentiable dynamical systems, and topological entrophy. Contains definitions and exercises for problem-solving practice.
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Basic Concepts and Facts of the Theory of Dynamical Systems
Dynamical Systems on Manifolds of Dimension 1 and 2
Transversality Theory and Generic Properties
5 other sections not shown
algebraic automorphism Anosov diffeomorphism assume Banach space Borel C1-close called chart circle class C1 closed orbit compact manifold condition Consequently Consider constant contains continuous map converges coordinates Corollary critical elements critical points defined Definition denote dense differential equations dimM dynamical system eigenvalues entropy equal equivalent Example Exercise expanding mappings follows function given Hence Hint homeomorphism hyperbolic fixed point iff there exists inequality integer integral curves intersects interval invariant measure Lebesgue measure Lemma Let q manifold and let matrix metric dynamical system metric space Morse-Smale non-wandering points obtain open set orientation periodic points phase portraits point x0 point xeM prime period probability measure Proof Let proof of Theorem properties prove Riemannian manifold rotation number satisfies sequence structurally stable submanifold subspace Suppose surjection tangent topological entropy topologically conjugate torus T2 trajectory transversal Tx(M uniquely ergodic unstable manifolds vector field write x0eM