scholarly journals INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES

2016 ◽  
Vol 4 ◽  
Author(s):  
NATHANAEL ACKERMAN ◽  
CAMERON FREER ◽  
REHANA PATEL
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Isomorphism Class ◽  
Invariant Measures ◽  
Pointwise Stabilizer

Let$L$be a countable language. We say that a countable infinite$L$-structure${\mathcal{M}}$admits an invariant measure when there is a probability measure on the space of$L$-structures with the same underlying set as${\mathcal{M}}$that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of${\mathcal{M}}$. We show that${\mathcal{M}}$admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in$\text{Aut}({\mathcal{M}})$of an arbitrary finite tuple of${\mathcal{M}}$fixes no additional points. When${\mathcal{M}}$is a Fraïssé limit in a relational language, this amounts to requiring that the age of${\mathcal{M}}$have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049
Author(s):  
A. BOYARSKY ◽  
Y. S. LOU
Keyword(s):  
Invariant Measure ◽  
Additional Condition ◽  
Invariant Measures ◽  
Chaotic Behavior ◽  
Finite Collection ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

scholarly journals On Relatively Invariant Measures

1960 ◽  
Vol 12 ◽  
pp. 367-373
Author(s):  
Mark Mahowald
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Locally Compact Groups ◽  
Major Portion ◽  
Compact Groups ◽  
Locally Compact ◽  
Multiplier Function ◽  
Baire Measure

In this note we will discuss the question of the measurability of the multiplier function of a relatively invariant measure on a group. That is, for a group G, σ-ring S, and a measure μ defined on the sets of S, we assume: E in S, x in G implies xE is in S and μ(XE) = σ(x)μ(E) and study the measurability of the function σ(x).The problem was discussed by Halmos (1, p. 265), on locally compact groups and there the situation proved to be as nice as it could be, that is, if the measure is a non-trivial, relatively invariant Baire measure then the multiplier function is continuous. We prove two theorems for groups in which no topology is assumed. In the first theorem we assume a shearing condition and answer the question completely. The second theorem places a condition on the measure and weakens the shearing assumption. Its proof is complicated and occupies the major portion of this paper.

Conditionally invariant measures for Anosov maps with small holes

1998 ◽  
Vol 18 (5) ◽  
pp. 1049-1073 ◽  
Author(s):  
N. CHERNOV ◽  
R. MARKARIAN ◽  
S. TROUBETZKOY
Keyword(s):  
Invariant Measure ◽  
Existence And Uniqueness ◽  
Invariant Measures ◽  
Smooth Boundary ◽  
Piecewise Smooth ◽  
Piecewise Smooth Boundary ◽  
Conditional Distributions ◽  
The Past ◽  
Set Of Points ◽  
Small Holes

We study Anosov diffeomorphisms on surfaces in which some small ‘holes’ are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories never enter holes under the past iterations of the map is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.

CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES

1996 ◽  
Vol 06 (06) ◽  
pp. 1143-1151
Author(s):  
E. A. SATAEV
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Expanding Mapping

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

CONTROLLING CHAOS AND THE INVERSE FROBENIUS–PERRON PROBLEM: GLOBAL STABILIZATION OF ARBITRARY INVARIANT MEASURES

2000 ◽  
Vol 10 (05) ◽  
pp. 1033-1050 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Stochastic Matrix ◽  
Open Loop ◽  
Global Stabilization ◽  
Invariant Density ◽  
Controlling Chaos

The inverse Frobenius–Perron problem (IFPP) is a global open-loop strategy to control chaos. The goal of our IFPP is to design a dynamical system in ℜn which is: (1) nearby the original dynamical system, and (2) has a desired invariant density. We reduce the question of stabilizing an arbitrary invariant measure, to the question of a hyperplane intersecting a unit hyperbox; several controllability theorems follow. We present a generalization of Baker maps with an arbitrary grammar and whose FP operator is the required stochastic matrix.

Invariant measures for Markov chains with no irreducibility assumptions

1988 ◽  
Vol 25 (A) ◽  
pp. 275-285 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Probability Theory ◽  
Invariant Measure ◽  
Markov Chains ◽  
Invariant Measures ◽  
Random Coefficient ◽  
General Function ◽  
Autoregressive Processes ◽  
Detailed Consideration ◽  
Positive Recurrence ◽  
Countable Space

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself. In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain. The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

Stochastic order for invariant measures of the contact process

2003 ◽  
Vol 40 (4) ◽  
pp. 1155-1157
Author(s):  
E. D. Andjel
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Contact Process ◽  
Stochastic Order

We show that the second lowest invariant measure of the contact process, introduced by Salzano and Schonmann (1997), is stochastically lower that any invariant measure which puts no mass on the empty configuration.

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