Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the Lebesgue measure l.
$T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any choice of a:
We've already showed, that $T_2$ is ergodic. $T_a$ is just ergodic if a is irrational. Now I still have to show, that the two systems are not isomorphic for irrational a. Don't know how to do this? I first tried to assume that there is an isomorphism and to construct a contradiction…I failed. What are other invariants of dynamical systems under isomorphism besides ergodicity? may someone help me?
There are a few possibilities here. I'm gonna use $m$ to refer to Lebesgue measure on the unit interval $[0 , 1)$. These are roughly in ascending order of "sophistication".
(1) The map $T_a$ is invertible, meaning that it not only has the property that $m(A) = m \left(T_a^{-1} A \right)$ for all measurable $A$, which is the definition of a measure-preserving map, but also $m(A) = m(T_a A)$ for all measurable $A$. The map $T_2$ does not have this property. Note that it's not quite enough to just observe that $T_2$ is not invertible as a map from $[0, 1)$ to itself. As others remarked, a lot can happen on a null set.
(2) The map $T_2$ is weakly mixing, strongly mixing, and even Bernoulli. The map $T_a$ is none of these.
(3) The map $T_2$ has positive entropy, whereas $T_a$ has entropy $0$.