A question on ergodic theory: topological mixing and invariant measures

In summary: So all you need to do is remove the first sentence in red, and your logic is correct.In summary, the conversation discusses a question on ergodic theory and how to prove that a measure invariant transformation cannot be mixing with respect to the probability measure in a compact metric space. The proof involves showing that for infinitely many natural numbers, the preimage of a nonempty open set under the transformation does not intersect with another open set. This contradicts the requirement for mixing and completes the proof.
  • #1
Alex V
1
0
Hi All,

This is a question on ergodic theory - not quite analysis, but as close as you can get to it, so I decided to post it here.

Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. Suppose also that $\mu(A) > 0$ for any nonempty set $A \subset X$. If we take a measure invariant transformation $T:X->X$ and assume it is NOT topologically mixing, how do we show it CANNOT be mixing with respect to $\mu$?

This is how I would attempt it. Take two sets nonempty open sets$A$ and $B$ in $X$.

Since we know that T is NOT topologically mixing, there are infinitely many natural numbers $n \in \mathbb{N}$ such that $T^{n}(A) \cap B = \emptyset$.

The preimage of $T^{n}(A)$ is $T^{n-1}(A)$. By measure preservation, we have $\mu (T^{n-1}(A)) = \mu(T^{n}(A))$. By repeated argument, we eventually have $\mu (T^{-n}(A)) = \mu(A)$.

Now we have $T^{-n}(A) \cap B = \emptyset$ for infinitely many $n$. Hence $\mu(T^{-n}(A) \cap B) = \emptyset$

This contradicts the requirement that for mixing, we need $\mu(T^{-n}(A) \cap B) = \mu(A)\mu(B)$ in the limit of $n$ tending to infinity, as the our assumption was that the measure of any nonempty set is bigger than zero.

Proof complete.

Is this correct, or are there any gaps or errors in my logic? If it is faulty, I'd be grateful to see the correct version. Thanks!
 
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  • #2
Alex V said:
Hi All,

This is a question on ergodic theory - not quite analysis, but as close as you can get to it, so I decided to post it here.

Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. Suppose also that $\mu(A) > 0$ for any nonempty set $A \subset X$. If we take a measure invariant transformation $T:X->X$ and assume it is NOT topologically mixing, how do we show it CANNOT be mixing with respect to $\mu$?

This is how I would attempt it. Take two sets nonempty open sets $\color{red}{A}$ and $\color{red}{B}$ in $\color{red}{X}$.

Since we know that T is NOT topologically mixing, there are infinitely many natural numbers $\color{red}{n \in \mathbb{N}}$ such that $\color{red}{T^{n}(A) \cap B = \emptyset}$.


The preimage of $T^{n}(A)$ is $T^{n-1}(A)$. By measure preservation, we have $\mu (T^{n-1}(A)) = \mu(T^{n}(A))$. By repeated argument, we eventually have $\mu (T^{-n}(A)) = \mu(A)$.

Now we have $T^{-n}(A) \cap B = \emptyset$ for infinitely many $n$. Hence $\mu(T^{-n}(A) \cap B) = \emptyset$

This contradicts the requirement that for mixing, we need $\mu(T^{-n}(A) \cap B) = \mu(A)\mu(B)$ in the limit of $n$ tending to infinity, as the our assumption was that the measure of any nonempty set is bigger than zero.

Proof complete.

Is this correct, or are there any gaps or errors in my logic? If it is faulty, I'd be grateful to see the correct version. Thanks!
Hi Alex, and welcome to MHB!

Your proof looks good. The only thing that needs adjusting is the part I have highlighted in red. The definition of topological mixing is that for all pairs of open sets $A$ and $B$, $T^{n}(A) \cap B \ne \emptyset$ for all sufficiently large $n$. If you want to negate a "for all" statement, the negation must always be a "there exists" statement. In this case, the definition of NOT topologically mixing is that there exist open sets $A$ and $B$ such that $T^{n}(A) \cap B = \emptyset$ for infinitely many $n$. In other words, you can't randomly choose $A$ and $B$ before mentioning the definition of not topologically mixing. The existence of such sets is actually included in that definition.
 

FAQ: A question on ergodic theory: topological mixing and invariant measures

What is ergodic theory and how does it relate to topological mixing and invariant measures?

Ergodic theory is a branch of mathematics that studies the behavior of dynamical systems, which are systems that change over time. Topological mixing and invariant measures are two concepts within ergodic theory that help to describe the long-term behavior of a dynamical system. Topological mixing refers to the idea that the system's states become increasingly mixed together as time goes on, while invariant measures are probability measures that remain unchanged by the system's dynamics.

What is topological mixing and why is it important in ergodic theory?

Topological mixing is a property of dynamical systems where the states become increasingly mixed together as time goes on. This is important in ergodic theory because it helps to characterize the long-term behavior of a system. Systems that exhibit topological mixing are considered to have chaotic behavior, meaning that they are highly sensitive to initial conditions and exhibit unpredictable behavior over time.

How are invariant measures related to topological mixing?

Invariant measures are probability measures that remain unchanged by the dynamics of a system. They provide a way to measure the long-term behavior of a system and are closely related to topological mixing. In systems that exhibit topological mixing, there is typically a unique invariant measure that describes the long-term behavior of the system.

Can a system exhibit topological mixing without having an invariant measure?

No, a system cannot exhibit topological mixing without having an invariant measure. Invariant measures are essential for characterizing the long-term behavior of a system, and without them, topological mixing cannot be properly defined.

How are topological mixing and invariant measures applied in real-world scenarios?

Topological mixing and invariant measures have applications in various fields, including physics, biology, and economics. They can be used to study complex systems and make predictions about their behavior over time. In particular, they have been applied in climate modeling, population dynamics, and financial market analysis.

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