Related papers: On NIP and invariant measures

We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the…

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of $\mathbb{R}$ which is Baire (=has the Baire property, BP) or measurable is shift-compact: it contains some subsequence of any null sequence to within translation by…

Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have…

We study some dynamical aspects of the action of automorphisms in model theory in particular in the presence of invariant measures. We give some characterizations for NIP theories in terms of dynamics of automorphisms and invariant measures…

We study existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. Existence is obtained from tightness of time averages under a weak regularity assumption inspired by…

We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global…

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from…

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the…

In this note we study when an invariant probability measure lifts to an invariant measure. Consider a standard Borel space $X$, a Borel probability measure $\mu$ on $X$, a Borel map $T \colon X \to X$ preserving $\mu$, a compact metric…

Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure 0. Under an extra technical hypothesis, this also holds if…

We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type…

Let $G$ be a countable discrete group with an orthogonal representation $\alpha$ on a real Hilbert space $H$. We prove $L_p$ Poincar\'e inequalities for the group measure space $L_\infty(\Omega_H,\gamma)\rtimes G$, where both the group…

We introduce several notions and give examples. We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for $l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD, where ${\rm Stab}( \langle \mathcal…

We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this…

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional…

We address the type I dichotomy for two-step nilpotent locally compact groups. Invoking work of Baggett-Kleppner, we characterize the closed points of the unitary dual of such a group $G$ purely in terms of the group structure. An algebraic…

In this paper we investigate the following questions. Let $\mu, \nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous…

A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$, either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist…

We introduce a family of local ranks DQ depending on a finite set Q of pairs of the form (\varphi(x,y),q(y)) where \varphi(x,y) is a formula and q(y) is a global type. We prove that in any NSOP1 theory these ranks satisfy some desirable…

We prove that a hypergroup admitting a countable basis and an invariant Haar measure has normed convergence property if and only if it is compact.