Ergodic transformation

Ergodic transformations or ergodic mapping are terms from probability theory and the theory of dynamic systems . Clearly, ergodicity of an image means that almost all points of the probability space lie in a single orbit of the dynamic system.

definition

Let it be a probability measure on a measuring space and a dimension-preserving mapping . ${\displaystyle \mu }$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle T\colon \Omega \to \Omega }$

Then an ergodic transformation, if and only if for every set that satisfies, is always either ${\displaystyle T}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle T^{-1}(A)=A}$

${\displaystyle \mu (A)=0\,{\text{ oder }}\,\mu (A)=1}$

applies. Here referred to the archetype of under . ${\displaystyle T^{-1}(A)}$ ${\displaystyle A}$ ${\displaystyle T}$

Other equivalent definitions can be given:

• In compact form, the above definition reads that the σ-algebra of T-invariant events should be a μ-trivial σ-algebra . ${\displaystyle {\mathcal {I}}}$
• Equivalent to this is that every measurable function is almost certainly constant. ${\displaystyle {\mathcal {I}}}$
• Alternatively, one can also demand that the only -invariant functions are the constant functions . A function is called -invariant if the equation applies to almost all of them . ${\displaystyle T}$ ${\displaystyle f\in L^{1}(\Omega ,\mu )}$ ${\displaystyle T}$ ${\displaystyle x\in \Omega }$ ${\displaystyle f(Tx)=f(x)}$

properties

• If is invertible, then: because all orbits ${\displaystyle T}$

${\displaystyle \left\{T^{n}x,n\in \mathbb {Z} \right\}}$

are -invariant (with ) an ergodic transformation , in particular exactly one orbit must have dimension 1 and all other orbits must have dimension 0. In particular, an invertible ergodic transformation defines an ergodic action of the group of integers . ${\displaystyle x\in \Omega }$ ${\displaystyle T}$ ${\displaystyle \mathbb {Z} }$

• Birkhoff's ergodic theorem applies to ergodic transformations :

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=0}^{n-1}f(T^{j}x)=\int _{\Omega }fd\mu }$

for almost everyone and every function . ${\displaystyle \mu }$ ${\displaystyle x\in \Omega }$ ${\displaystyle f\in L^{1}(\Omega ,\mu )}$

Examples

• The Lebesgue measure is an ergodic measure for the angle doubling mapping . ${\displaystyle T\colon \left[0,1\right]\to \left[0,1\right]}$
• The 2-dimensional Lebesgue measure is an ergodic measure for the baker's transformation . ${\displaystyle T\colon \left[0,1\right]^{2}\to \left[0,1\right]^{2}}$

literature

• A. Katok and B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995. ISBN 0-521-34187-6
• B. Bekka and M. Mayer: Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces. London Math. Soc. Lec. Notes # 269. Cambridge U. Press, Cambridge, 2000. ISBN 0-521-66030-0

Web links

• C.Walkden: Ergodic Theory (Chapter 5)