Følner sequence

Definition

A Følner sequence is a sequence of finite subsets of a group that asymptotically approximate the entire group in terms of size and symmetry. These sequences are crucial in understanding amenable groups, as they help analyze the behavior of functions and averages over the group under the action of its elements. The properties of Følner sequences allow for the application of the mean and pointwise ergodic theorems, which are foundational in ergodic theory.

5 Must Know Facts For Your Next Test

1. Følner sequences are designed such that the boundary size grows slower than the size of the sets themselves, ensuring that their averages converge appropriately.
2. In an amenable group, every Følner sequence leads to a unique limit point in terms of average values, facilitating the application of ergodic theorems.
3. The existence of a Følner sequence is equivalent to a group being amenable, making them central to the study of such groups.
4. Følner sequences play a critical role in establishing results like the mean ergodic theorem, which relates time averages to space averages in an amenable group.
5. The use of Følner sequences allows one to derive pointwise convergence results for functions over groups, thus linking local behavior with global properties.

Review Questions

• How do Følner sequences demonstrate the properties of amenable groups?
  • Følner sequences showcase amenable groups by providing a way to approximate group actions with finite sets whose boundaries grow slower than their sizes. This characteristic ensures that averages calculated over these sequences converge to a limit. Hence, they serve as a tool to illustrate why certain groups can support invariant means and satisfy ergodic properties.
• What role do Følner sequences play in proving the mean ergodic theorem for amenable groups?
  • Følner sequences are integral in proving the mean ergodic theorem for amenable groups because they help establish a connection between time averages and space averages. By employing these sequences, one can show that as one takes larger sets in the sequence, the average values converge to an invariant mean. This convergence is critical to demonstrating that long-term behaviors can be accurately predicted through these finite approximations.
• Critically analyze how the properties of Følner sequences influence pointwise convergence results for functions defined on amenable groups.
  • The properties of Følner sequences significantly impact pointwise convergence results by ensuring that for functions defined on amenable groups, their averages converge at almost every point. Since these sequences capture the essence of how functions behave under group actions while controlling boundary effects, they allow for robust conclusions about the limiting behavior. The ability to link local function behavior with global properties highlights the importance of Følner sequences in understanding ergodic phenomena.

Related terms

Amenable groups: Groups for which every Følner sequence has a non-empty limit set, allowing for the existence of invariant means on bounded functions.

Ergodic theorem: Theorems that describe the long-term average behavior of dynamical systems, often applied to study the properties of measures in relation to group actions.

Invariant mean: A linear functional on the space of bounded functions that remains unchanged under the action of the group, reflecting the averaging process over a Følner sequence.