Statistical stability in dynamical systems

Mike Todd (University of St Andrews)

Thursday 14th March 16:00-17:00 Maths 311B

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 (UCD).

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