scholarly journals Dynamical simplices and Fraïssé theory

2018 ◽  
Vol 39 (11) ◽  
pp. 3111-3126 ◽  
Author(s):  
JULIEN MELLERAY
Keyword(s):  
Invariant Measures ◽  
Probability Measures ◽  
Choquet Simplex ◽  
Cantor Space ◽  
Orbit Equivalent ◽  
Minimal Homeomorphisms

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$ . We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.

scholarly journals Lebesgue orbit equivalence of multidimensional Borel flows: A picturebook of tilings

2016 ◽  
Vol 37 (6) ◽  
pp. 1966-1996
Author(s):  
KONSTANTIN SLUTSKY
Keyword(s):  
Lebesgue Measure ◽  
Probability Measures ◽  
Orbit Equivalence ◽  
Orbit Equivalent

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue orbit equivalence, by which we mean an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non-smooth $\mathbb{R}^{d}$-flows are shown to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

scholarly journals Order-Invariant Measures on Fixed Causal Sets

2012 ◽  
Vol 21 (3) ◽  
pp. 330-357 ◽  
Author(s):  
GRAHAM BRIGHTWELL ◽  
MALWINA LUCZAK
Keyword(s):  
Linear Extension ◽  
Invariant Measures ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Order Type ◽  
Probability Measures ◽  
Causal Sets ◽  
Natural Extensions ◽  
Countably Infinite ◽  
Order Invariant

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

scholarly journals Full groups of minimal homeomorphisms and Baire category methods

2014 ◽  
Vol 36 (2) ◽  
pp. 550-573
Author(s):  
TOMÁS IBARLUCÍA ◽  
JULIEN MELLERAY
Keyword(s):  
Baire Category ◽  
Full Group ◽  
Group Topology ◽  
Cantor Space ◽  
Polish Group ◽  
Minimal Homeomorphisms

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space$X$, showing that these groups do not admit a compatible Polish group topology and, in the case of$\mathbb{Z}$-actions, are coanalytic non-Borel inside$\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside$\text{Homeo}(X)$.

Entropy and rotation sets: A toy model approach

2016 ◽  
Vol 18 (05) ◽  
pp. 1550083 ◽  
Author(s):  
Tamara Kucherenko ◽  
Christian Wolf
Keyword(s):  
Invariant Measures ◽  
Maximal Entropy ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Lipschitz Continuous ◽  
Exposed Point ◽  
Continuous Potential ◽  
Rotation Set ◽  
Measures Of Maximal Entropy ◽  
Model Approach

Given a continuous dynamical system [Formula: see text] on a compact metric space [Formula: see text] and a continuous potential [Formula: see text], the generalized rotation set is the subset of [Formula: see text] consisting of all integrals of [Formula: see text] with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

scholarly journals Positive entropy invariant measures on the space of lattices with escape of mass

2011 ◽  
Vol 32 (1) ◽  
pp. 141-157 ◽  
Author(s):  
SHIRALI KADYROV
Keyword(s):  
Invariant Measures ◽  
Diagonal Element ◽  
Probability Measures ◽  
Positive Entropy ◽  
Limit Measure ◽  
High Entropy

AbstractOn the space of unimodular lattices, we construct a sequence of invariant probability measures under a singular diagonal element with high entropy, and show that the limit measure is zero.

Automorphism–invariant measures on ℵ0-categorical structures without the independence property

1996 ◽  
Vol 61 (2) ◽  
pp. 640-652
Author(s):  
Douglas E. Ensley
Keyword(s):  
Boolean Algebra ◽  
Invariant Measures ◽  
Measure Zero ◽  
Probability Measures ◽  
Independence Property ◽  
Natural Action ◽  
Zero Sets ◽  
Finitely Additive Probability ◽  
Additive Probability

AbstractWe address the classification of the possible finitely-additive probability measures on the Boolean algebra of definable subsets of M which are invariant under the natural action of Aut(M). This pursuit requires a generalization of Shelah's forking formulas [8] to “essentially measure zero” sets and an application of Myer's “rank diagram” [5] of the Boolean algebra under consideration. The classification is completed for a large class of ℵ0-categorical structures without the independence property including those which are stable.

scholarly journals Invariant measures on stationary Bratteli diagrams

2009 ◽  
Vol 30 (4) ◽  
pp. 973-1007 ◽  
Author(s):  
S. BEZUGLYI ◽  
J. KWIATKOWSKI ◽  
K. MEDYNETS ◽  
B. SOLOMYAK
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Complex Number ◽  
Incidence Matrix ◽  
Invariant Measures ◽  
Explicit Description ◽  
Bratteli Diagram ◽  
Probability Measures ◽  
Bratteli Diagrams ◽  
Substitution Dynamical Systems

AbstractWe study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Sur Une Équation Fonctionnelle Et SES Applications: Une Extension Du Théorème De Kesten-Stigum Concernant Des Processus De Branchement

1997 ◽  
Vol 29 (2) ◽  
pp. 353-373 ◽  
Author(s):  
Quansheng Liu
Keyword(s):  
Invariant Measures ◽  
Random Variables ◽  
Particle Systems ◽  
Independent Random Variables ◽  
Probability Measures ◽  
Optimal Conditions ◽  
Branching Random Walks ◽  
Random Integer ◽  
Watson Process ◽  
Infinite Particle

Given a random integer N ≧ 0 and a sequence of random variables Ai ≧ 0, we define a transformation T on the class of probability measures on [0, ∞) by letting Tμ be the distribution of , where {Zi} are independent random variables with distribution μ, which are independent of N and of {Ai} as well. We obtain the optimal conditions for the functional equation μ = Tμ to have a non-trivial solution of finite mean, and we study the existence of moments of the solutions. The work unifies the corresponding theorems of Kesten-Stigum concerning the Galton-Watson process, of Biggins for branching random walks, of Kahane-Peyrière for a model of turbulence of Yaglom made precise by Mandelbrot, and of Durrett-Liggett for the study of invariant measures for certain infinite particle systems.

Sign in / Sign up

Export Citation Format

Share Document