Category Archives: smooth dynamics

Dynamics of smooth maps on manifolds

Gibbs measures have local product structure

Let be a compact smooth manifold and a transitive Anosov diffeomorphism. If is an -invariant Borel probability measure on that is absolutely continuous with respect to volume, then the Hopf argument can be used to show that is ergodic. In … Continue reading

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Fubini foiled

An important issue in hyperbolic dynamics is that of absolute continuity. Suppose some neighbourhood of a smooth manifold is foliated by a collection of smooth submanifolds , where is some indexing set. (Here “smooth” may mean , or , or … Continue reading

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The stable manifold theorem

The stable manifold theorem is one of the most important in the theory of non-linear ODEs and dynamical systems. Unfortunately, some of the standard introductory texts (Hirsch–Smale, Perko) either do not give a proof, or do not motivate the proof, … Continue reading

Posted in ODE, smooth dynamics, theorems | Tagged , | 3 Comments