Department of Mathematics

Open Problems
in
Infinite Ergodic Theory

There are many more open problems then those listed below. We are only listing (at this time) the ones relating to our own research.

If you are interested in any of these problems, let us know and we will put you in contact with others interested in the same.

Standing Assumptions: Unless otherwise indicated, all transformations will be Ergodic, Invertible, and Measure Preserving on a Non-Atomic, Sigma-Finite Lebesgue Space of Infinite Measure.

Stanley Eigen
Arshag Hajian

Definitions and Notations

Exhaustive Weakly Wandering

  • \( (X, \mathscr{B}, \mu , T) \) denotes an ergodic, invertible, infinite measure preserving transformations on a \( \sigma \)-finite (Lebesgue) measure space.
  • \( \mathbb{A} \) is an infinite sequence (subset) of the integers.
  • \( W \) is a (measurable) subset of \( X \) $$ X = \dot\cup_{a \in \mathbb{A}} T^a W \ (disjoint)$$
  • The Set \( W \) and sequence \( \mathbb{A} \) are called Exhaustive Weakly Wandering (EWW) set and sequence respectively.

Explicit Exhaustive Weakly Wandering Sets and Sequences

    (a) No exhaustive weakly wandering sets or sequences are known for any of the following examples.
    (b) It is unknown if any of the following are of finite type.
    (c) It is unknown if any of the following can commute with a non measure preserving transformation.
    1. Geodesic Flows on Hyperbolic Surfaces
    2. Markov maps
    3. Hopf's map
    4. Maharam Transformations
    5. Kakutani-Parry Transformations (ergodic k-index)
    6. Inner Automorphism
    7. Random Walks on the Integers

Weakly Wandering Sequences

  1. Does the collection of all Weakly Wandering Sequences completely define the transformation?
  2. Related to the previous question: Let WW2(T) denote the collection of Two-Sided weakly wandering sequences for T. Let tau denote the Shift-map on WW2(T). Can a measure be defined on WW2(T) in some significant manner, which is invariant for the map tau?

Exhaustive Weakly Wandering Sets

  1. For a fixed transformation T, what are the possible sizes of the the exhaustive weakly wandering sets? For example, there exist transformations for which the size of the exhaustive weakly wandering sets can only be infinite.
  2. As above, can the size be bounded away from 0 but include sets of finite measure?

Dissipative Sequences

  1. The only known method to construct dissipative sequences is by using weakly wandering sequences. Find a direct method to construct dissipative sequences.
  2. Given a transformation find a characterizing property for its dissipative sequences. A good example to start with is the classic random walk on the integers.

Positive-Type

  1. What are the possible Set values for Alpha? Examples are known for where Alpha is a single value in the range (0,1], and where Alpha is an interval of the form (0,alpha]. There are no known examples where Alpha takes exactly two values. There are no known examples where Alpha is an interval bounded away from 0. A good place to start would be with a Rank-Two construction.

Harder?

  1. If the only transformations that commute with T are measure preserving, is T of finite type?
  2. Can any finite measure preserving ergodic transformation occur as the induced transformation on an exhaustive weakly wandering set.
  3. If T x S is completely dissipative how are the exhaustive weakly wandering sequences of T and S related?
  4. If T and S are Similar, how are the exhaustive weakly wandering sequences related?