Generic Stationary Measures and Actions

Let $G$ be a countably infinite group, and let $\mu$ be a generating probability measure on $G$. We study the space of $\mu$-stationary Borel probability measures on a topological $G$ space, and in particular on $Z^G$, where $Z$ is any perfect Polish space. We also study the space of $\mu$-stationary, measurable $G$-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When $\mu$ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $(G,\mu)$. When $Z$ is compact, this implies that the simplex of $\mu$-stationary measures on $Z^G$ is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on $\{0,1\}^G$. We furthermore show that if the action of $G$ on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when $G$ has property (T), the ergodic actions are meager. We also construct a group $G$ without property (T) such that the ergodic actions are not dense, for some $\mu$. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual.Comment: To appear in the Transactions of the AMS, 49 page

Property (T) and the Furstenberg Entropy of Nonsingular Actions

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condition

Generic Stationary Measures and Actions

Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on Z^G, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When μ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G, μ). When Z is compact, this implies that the simplex of μ-stationary measures on Z^G is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on {0, 1}^G. We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G has property (T), the ergodic actions are meager. We also construct a group G without property (T) such that the ergodic actions are not dense, for some μ. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual

Property (T) and the Furstenberg Entropy of Nonsingular Actions

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condition

Generic Stationary Measures and Actions

Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on Z^G, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When μ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G, μ). When Z is compact, this implies that the simplex of μ-stationary measures on Z^G is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on {0, 1}^G. We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G has property (T), the ergodic actions are meager. We also construct a group G without property (T) such that the ergodic actions are not dense, for some μ. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual

Efficacy and mechanisms underlying a gamified attention bias modification training in anxious youth: protocol for a randomized controlled trial

Background Attention bias modification training (ABMT) and cognitive behavioral therapy (CBT) likely target different aspects of aberrant threat responses in anxiety disorders and may be combined to maximize therapeutic benefit. However, studies investigating the effect of ABMT in the context of CBT have yielded mixed results. Here, we propose an enhanced ABMT to target the attentional bias towards threat, in addition to classic CBT for anxiety disorders in youth. This enhanced ABMT integrates the modified dot-probe task used in previous studies, where a target is always presented at the previous location of the neutral and not the simultaneously presented threatening stimulus, with a visual search, where the targets are always presented distally of threatening distractors. These two training elements (modified dot-probe and visual search) are embedded in an engaging game to foster motivation and adherence. Our goal is to determine the efficacy of the enhanced ABMT in the context of CBT. Further, we aim to replicate two previous findings: (a) aberrant amygdala connectivity being the neurobiological correlate of the attentional bias towards threat at baseline; and (b) amygdala connectivity being a mediator of the ABMT effect. We will also explore moderators of treatment response (age, sex, depressive symptoms and irritability) on a behavioral and neuronal level. Methods One hundred and twenty youth (8–17 years old) with a primary anxiety disorder diagnosis all receive CBT and are randomized to nine weeks of either active or control ABMT and symptom improvement will be compared between the two study arms. We will also recruit 60 healthy comparison youth, who along with eligible anxious youth, will be assessed with the dot-probe task during fMRI (anxious youth: before and after training; healthy volunteers: second measurement twelve weeks after initial assessment). Discussion The present study will contribute to the literature by (1) potentially replicating that aberrant amygdala connectivity mediates the attentional bias towards threat in anxious youth; (2) determining the efficacy of enhanced ABMT; and (3) advancing our understanding of the mechanisms underlying ABMT. Trial registration Clinicaltrials.gov: NCT03283930 Trial registration date: September 14th 2017. The trial registration took place retrospectively. Data acquisition started February 1st 2017

An integrative genomics approach identifies Hypoxia Inducible Factor-1 (HIF-1)-target genes that form the core response to hypoxia

The transcription factor Hypoxia-inducible factor 1 (HIF-1) plays a central role in the transcriptional response to oxygen flux. To gain insight into the molecular pathways regulated by HIF-1, it is essential to identify the downstream-target genes. We report here a strategy to identify HIF-1-target genes based on an integrative genomic approach combining computational strategies and experimental validation. To identify HIF-1-target genes microarrays data sets were used to rank genes based on their differential response to hypoxia. The proximal promoters of these genes were then analyzed for the presence of conserved HIF-1-binding sites. Genes were scored and ranked based on their response to hypoxia and their HIF-binding site score. Using this strategy we recovered 41% of the previously confirmed HIF-1-target genes that responded to hypoxia in the microarrays and provide a catalogue of predicted HIF-1 targets. We present experimental validation for ANKRD37 as a novel HIF-1-target gene. Together these analyses demonstrate the potential to recover novel HIF-1-target genes and the discovery of mammalian-regulatory elements operative in the context of microarray data sets