Abstract
We explore the concept of metastability or almost-invariance in open dynamical systems. In such systems, the loss of mass through a ``hole'' occurs in the presence of metastability. We extend existing techniques for finding almost-invariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.
Highlights
The classical setting for dynamical systems is one in which a transformation T : X → X is iterated to generate forward trajectories of infinite length
For the α-shift example we find that the invariance ratio for the closed system is always greater than that for the open system for this particular class of intervals
While considerable research has been done on almost-invariance in closed systems, little, if any attention has been paid to open systems, where trajectories terminate when entering a hole, and the time-asymptotic dynamics occurs on a survivor set
Summary
The classical setting for dynamical systems is one in which a transformation T : X → X is iterated to generate forward trajectories of infinite length. We will call a dynamical system open if its domain contains holes and closed if not. Our focus is on systems with some (piecewise) smoothness and their ergodic properties. There has been a burgeoning interest in the mathematical characterisation of these types of open systems [36, 17, 34, 32, 7]; see the survey article [18] and references therein. Throughout we will assume that X is a smooth Riemannian manifold embedded in Rd and T a piecewise smooth transformation on X
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