Transience in dynamical systems
Time and place
1 PM on Thursday, December 2nd, 2010; NAC 6-113
Mike Todd (Boston University)
Abstract
The statistical behavior of a dynamical system $f:X \rightarrow X$ can be understood through the set of invariant measures on that system. The interesting measures can be produced by taking a potential $\phi:X \rightarrow \mathbb{R}$ and finding a measure $\mu_\phi$ which maximizes thermodynamic quantities with respect to this potential - these are often Gibbs states. Gibbs states tend to be well behaved, having exponential decay of correlations, exponential large deviations etc.
On the other hand, a system $(X,f, \phi)$ may be transient: for such systems, the relevant measure, if it exists, is not a Gibbs state and does not exhibit exponential statistical properties. This situation has been observed for very natural potentials in some non-uniformly hyperbolic dynamical systems: for example the Manneville-Pomeau map, certain unimodal maps and certain countable Markov shifts. I will discuss such examples, showing how the onset of transience reflects the statistics of the system.