Abstract
A compact topologically mixing uniformly hyperbolic attractor for a diffeomorphism of a finite-dimensional manifold with Hölder continuous derivative carries a unique probability measure describing the long-time statistics of the forward orbits of almost all initial conditions in its basin. This was proved [28] by converting the orbits of the dynamical system into symbol sequences from a finite alphabet (indexed by time) and considering them as states of a statistical mechanical spin chain with a certain interaction energy which decays exponentially with separation, so leading to a unique phase. The results have been extended to many classes of non-uniformly hyperbolic system by different approaches (e.g. [30]).
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MacKay, R. Indecomposable Coupled Map Lattices with Non–unique Phase. In: Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Lecture Notes in Physics, vol 671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11360810_4
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