# Ergodic theory and Set Theory

This page gives information about one-day conference on Ergodic theory and set theory, the fourth in the series Set theory and its neighbours, which took place on Wednesday, 15th September 1999 at the London Mathematical Society building, De Morgan House, 57-58 Russell Square, London WC1.

The next meeting in the series, on Topology and set theory, will take place on Wednesday, 5th January, 2000.

The speakers at the meeting were

Here is the list of participants.

What are the connections between ergodic theory and set theory?

Below is an extremely telegraphic list of some theorems from the last few years showing some of the links. These theorems mostly illustrate the influence of (descriptive) set theory on various problems in ergodic theory.

Let G be the group of invertible measure preserving transformations on X a standard measure space. Then (Halmos) G is a Polish group. The ergodic transforamtions are a dense G_{\delta} set in G (Halmos) while the strongly mixing maps are a meagre F_{\sigma\delta} subset of G (Rohklin). The Bernoulli shifts are a \Pi^0_3 set in G. The entropy function is a Borel function, and (Orenstein) two Bernoulli shifts are isomorphic iff they have the same entropy, so that the conjugacy realtion on Bernoulli shifts is smooth.

The isomorphism relation between ergodic discrete spectrum transformations is inter-reducible to the identity relation on countable sets of real numbers (Foreman/Louveau, using Halmos-von-Neumann in one direction). E_0, (the equivalence relation of equal modulo finite difference on sets of natural numbers, or equivalently of equality mod the rationals on the reals) is Borel reducible to the conjugacy relation on property K automorphisms of any fixed entropy (Orenstein-Shields, Feldman), and so they are, eg, not smoothly classifiable. The conjugacy relation on normal transformations of a bounded order \alpha is Borel (Foreman), but the conjugacy relation on all measure preserving transformations is complete \Sigma^1_1 (Hjorth). There is a "turbulent" group action on a Polish space such that the corresponding equivalence relation is Borel reducible to norm 2 flows on T^2. So there is no classification of norm 2 distal flows by countable structures (Hjorth). The classs of measure distal transformations is a complete \Pi^1_1 set, and the function that associates to each transformation the least ordinal in an approximating tower is a \Pi^1_1 norm. In particular every countable ordinal occurs as the least ordinal of an approximating tower to some measure distal transformation and the m.d.t.s of order bounded by \alpha are a Borel set (Beleznay-Foreman).

More information about such issues will, we believe, be found in a forthcoming expository paper by Matt Foremann, and in a volume of papers, edited by Foreman and also forthcoming. (We don't have any publication/preprint details but you might like to try Foreman's website over the next month or two to see if some pointers to such details appear.)

In the opposite direction) has been the use of notions from ergodic theory in Hjorth et al's work on classifying equivalence relations. See, for example, Hjorth's Classification and orbit equivalence relations, and his talk at the first meeting in the Set theory and its neighbours series. The paper Borel equivalence relations and classifications of countable models, by Hjorth and Kechris, Ann. Pure and Appl. Logic 82 (1996), 221-272, gives an illustration of both tendencies.

Combinatorics and set theory (the third meeting in the series), including slides from the talks and related preprints.
Finite model theory and set theory (the second meeting in the series), including slides from the talks and related preprints.
Set theory, analysis and their neighbours (the first meeting in the series), inlcuding slides from the talks and related preprints.

We hope to keep the meetings fairly relaxed, allowing plenty of opportunity for informal discussion. We welcome and encourage anyone to participate. Please do tell anyone about the meeting who you think may be interested in it. And let us know if you would like to speak or have ideas for speakers at future meetings.

We would be grateful if you could email us to let us know if you intend to come, so that we can get a reasonable idea about number of people likely to attend. Nevertheless you are very welcome simply to turn up on the day if you make a late decision.

We have some limited funds to subsidise the travel expenses of graduate students who would like to attend. Please contact us for details.

We are very grateful to the LMS for allowing us to use De Morgan House as a venue and for their financial support for these meetings. De Morgan House is in the bottom left (i.e. south-east) corner of Russell Square, itself in the bottom left hand corner of this map of the area. The nearest tube station is Russell Square, but De Morgan House is also only a short walk from Euston, Euston Square and Goodge Street stations.

We are very grateful to the British Logic Colloquium for financial support for this meeting.

Mirna Džamonja and Charles Morgan

Last updated on 20th October 1999