Ergodicity

of a dynamical system

2020 Mathematics Subject Classification: Primary: 37A25 [MSN][ZBL]

A property considered in ergodic theory. Originally it was defined for a cascade $\{T^k\}$ or a flow (continuous-time dynamical system) $\{T_t\}$ with a finite invariant measure $\mu$ in the following way: If a function $f\in L_1(W,\mu)$ is given on the phase space $W$, then for almost-every point $w$ the time average along the trajectory of this point exists, that is,

$$\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f(T^kw),$$

or

$$\lim_{T\to\infty}\frac1T\int\limits_0^Tf(T_tw)dt,$$

exists, and coincides with the space average (that is, with $(\int fd\mu)/\mu(W)$). In this case one also speaks of ergodicity of $\mu$. In particular, for any measurable set $A\subset W$ the average time of a trajectory staying in $A$ is proportional to $\mu(A)$ for almost-every point (in fact, this property is equivalent to ergodicity). When the Birkhoff ergodic theorem had been proved it became clear that ergodicity is equivalent to metric transitivity. Therefore one spoke of ergodicity, meaning metric transitivity, in the more general situation when it was no longer suitable to talk of the equality of time and space averages (systems with an infinite invariant or quasi-invariant measure, not only flows and cascades, but also more general transformation groups and semi-groups).