Difference between revisions of "Strong ergodicity"
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''of a [[Topological dynamical system|topological dynamical system]] in the narrow sense (a flow or a cascade)'' | ''of a [[Topological dynamical system|topological dynamical system]] in the narrow sense (a flow or a cascade)'' | ||
| − | A property examined in [[Ergodic theory|ergodic theory]]. It consists of the following: 1) the system has a unique, invariant, normalized, regular, Borel measure | + | A property examined in [[Ergodic theory|ergodic theory]]. It consists of the following: 1) the system has a unique, invariant, normalized, regular, Borel measure $\mu$; 2) $\mu(U)>0$ for any non-empty open set $U$; and 3) for any bounded continuous function $f$ its time averages along any trajectory tend to $\int fd\mu$. |
| − | Although this definition holds independently of any restrictions on the phase space | + | Although this definition holds independently of any restrictions on the phase space $W$, it is used in practice when $W$ is a complete separable metric space (usually even a metric compactum). In this case strong ergodicity means that $W$ is an [[Ergodic set|ergodic set]]. Strong ergodicity implies that $W$ is a [[Minimal set|minimal set]] (but not vice-versa). Any ergodic flow or cascade in a [[Lebesgue space|Lebesgue space]] is metrically isomorphic to a topological flow or cascade with the property of strong ergodicity. |
Strong ergodicity is sometimes taken to mean only property 1); in this sense the term unique ergodicity is also used. | Strong ergodicity is sometimes taken to mean only property 1); in this sense the term unique ergodicity is also used. | ||
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====Comments==== | ====Comments==== | ||
| − | In Western literature the usual term is strict ergodicity. In the case that the phase space | + | In Western literature the usual term is strict ergodicity. In the case that the phase space $W$ is compact, the conditions 1) and 3) above are equivalent, so that in that case 3) characterizes unique ergodicity. If these conditions are fulfilled, then the support of the unique invariant normalized measure is the unique minimal subset of $W$. Thus, a topological dynamical system with a compact phase space is strictly ergodic if and only if it is uniquely ergodic and minimal. |
The result mentioned above that any ergodic flow or cascade in a Lebesgue space can be represented as a strictly ergodic flow or cascade has to be understood in the sense that the original system is not required to be topological (the transformations are only required to be measurable). In the case of an ergodic cascade with finite entropy, the representing system may be taken to be a closed, invariant, strictly ergodic subset of a shift system (or topological Bernoulli automorphism; see [[Symbolic dynamics|Symbolic dynamics]]). This result is known as the Jewett–Krieger theorem. | The result mentioned above that any ergodic flow or cascade in a Lebesgue space can be represented as a strictly ergodic flow or cascade has to be understood in the sense that the original system is not required to be topological (the transformations are only required to be measurable). In the case of an ergodic cascade with finite entropy, the representing system may be taken to be a closed, invariant, strictly ergodic subset of a shift system (or topological Bernoulli automorphism; see [[Symbolic dynamics|Symbolic dynamics]]). This result is known as the Jewett–Krieger theorem. | ||
Latest revision as of 17:35, 15 September 2014
of a topological dynamical system in the narrow sense (a flow or a cascade)
A property examined in ergodic theory. It consists of the following: 1) the system has a unique, invariant, normalized, regular, Borel measure $\mu$; 2) $\mu(U)>0$ for any non-empty open set $U$; and 3) for any bounded continuous function $f$ its time averages along any trajectory tend to $\int fd\mu$.
Although this definition holds independently of any restrictions on the phase space $W$, it is used in practice when $W$ is a complete separable metric space (usually even a metric compactum). In this case strong ergodicity means that $W$ is an ergodic set. Strong ergodicity implies that $W$ is a minimal set (but not vice-versa). Any ergodic flow or cascade in a Lebesgue space is metrically isomorphic to a topological flow or cascade with the property of strong ergodicity.
Strong ergodicity is sometimes taken to mean only property 1); in this sense the term unique ergodicity is also used.
Comments
In Western literature the usual term is strict ergodicity. In the case that the phase space $W$ is compact, the conditions 1) and 3) above are equivalent, so that in that case 3) characterizes unique ergodicity. If these conditions are fulfilled, then the support of the unique invariant normalized measure is the unique minimal subset of $W$. Thus, a topological dynamical system with a compact phase space is strictly ergodic if and only if it is uniquely ergodic and minimal.
The result mentioned above that any ergodic flow or cascade in a Lebesgue space can be represented as a strictly ergodic flow or cascade has to be understood in the sense that the original system is not required to be topological (the transformations are only required to be measurable). In the case of an ergodic cascade with finite entropy, the representing system may be taken to be a closed, invariant, strictly ergodic subset of a shift system (or topological Bernoulli automorphism; see Symbolic dynamics). This result is known as the Jewett–Krieger theorem.
References
| [a1] | K. Petersen, "Ergodic theory" , Cambridge Univ. Press (1983) |
Strong ergodicity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_ergodicity&oldid=13945