On minimal flows and definable amenability in some distal NIP theories

Distal and non-distal NIP theories

We study one way in which stable phenomena can exist in an NIP theory. We start by defining a notion of ‘pure instability’ that we call ‘distality’ in which no such phenomenon occurs. O-minimal theories and the p-adics for example are distal. Next, we try to understand what happens when distality fails. Given a type p over a sufficiently saturated model, we extract, in some sense, the stable pa...

Definable and Invariant Types in Enrichments of NIP Theories

NIP for some pair-like theories

Generalising work from [2] and [6], we give sufficient conditions for a theory TP to inherit NIP from T , where TP is an expansion of the theory T by a unary predicate P . We apply our result to theories, studied in [1], of the real field with a subgroup of the unit circle.

Definable Structures in O-minimal Theories: One Dimensional Types

Let N be a structure definable in an o-minimal structureM and p ∈ SN (N), a complete N -1-type. If dimM(p) = 1 then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N -definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to...

Type decomposition in NIP theories

A first order theory is NIP if all definable families of subsets have finite VCdimension. We provide a justification for the intuition that NIP structures should be a combination of stable and order-like components. More precisely, we prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and an order-like quotient.