Abstract:
If each member of a continuous family $(f_t)_t$ of dynamical systems possesses a `physical’ measure $\mu_t$ (that is, a measure describing the behaviour of Lebesgue-typical points), one can ask if the family of measures $(\mu_t)_t$ is also continuous in $t$: this is statistical stability, so called because the statistics (for example, in terms of Birkhoff averages for $(f_t, \mu_t)$) change continuously in $t$. I’ll discuss this problem for interval maps (eg tent maps, quadratic maps). Statistical stability can be destroyed by topological obstructions, or by a lack of uniform hyperbolicity. I’ll outline a general theory which guarantees statistical stability, giving examples to show the sharpness of our results. This is joint work with Neil Dobbs (UC Dublin).