Some results on the geometry of Choquet simplices

In this paper, we give some properties of Choquet simplices which lead to the characterization of the closed Choquet simplices of ⁿ.The results of this paper were presented at the Conference on Convexity and Foundation of Geometry held in Haifa (10–14 March 1975).     

Baire functions and spaces of Baire functions

The paper considers (1) the tightness of spaces of Baire functions and their subspaces endowed with the topology of pointwise convergence; (2) Z _σ-mappings of K-analytic spaces; (3) K _σ-analytic spaces (Tychonoff spaces that are Z _σ-images of K-analytic spaces). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003.     

Choquet simplices as spaces of invariant probability measures on post-critical sets

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak^∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f   is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential −ln|f^′|$-\ln |f^{\prime }|$.     

The Ramsey property for Banach spaces and Choquet simplices,and applications

We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group $\mathrm{Aut}(\mathbb{G})$ of surjective linear isometries of the Gurarij space $\mathbb{G}$ is extremely amenable, and that the canonical action $\mathrm{Aut}(\mathbb{P})?\mathbb{P}$ is the universal minimal flow of the group $\mathrm{Aut}(\mathbb{P})$ of affine homeomorphisms of the Poulsen simplex $\mathbb{P}$. This answers questions of Melleray–Tsankov and Conley–Törnquist.     

Gauges of Baire class one functions

Let K be a compact metric space and be a bounded Baire class one function. We proved that for any ε>0 there exists an upper semicontinuous positive function δ of f with finite oscillation index and |f(x)−f(y)|<ε whenever d(x,y)<min{δ(x),δ(y)}.     

Distances to spaces of Baire one functions

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps to estimate the distance of f to the space B ₁(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stating that in the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa’s result that states that whenever X is metric any weakly continuous function belongs to B ₁(X, E): our result here says that for an arbitrary we have

Affine functions on simplexes and extreme operators

IfK a simplex andX a Banach space thenA(K, X) denotes the space of affine continuous functions fromK toX with the supremum norm. The extreme points of the closed unit ball ofA(K, X) are characterized,X being supposed to satisfy certain conditions. This characterization is used to investigate the extreme compact operators from a Banach spaceX to the spaceA(K)=A(K, (− ∞, ∞)). This note is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank them for their helpful advice and kind encouragement.     

Affine similarity of refinable functions

In this paper, we consider certain affine similarity of refinable functions and establish, certain concection between some local and global properties of refinable functions, such as local and global linear independence, local smoothness and B-spline, local and global H?lder continuity.