scholarly journals Unique ergodicity for non-uniquely ergodic horocycle flows

2006 ◽  
Vol 16 (2) ◽  
pp. 411-433 ◽  
Author(s):  
François Ledrappier ◽  
◽  
Omri Sarig ◽  
Keyword(s):  
Unique Ergodicity ◽  
Uniquely Ergodic

On Certain K-Groups Associated with Minimal Flows

1998 ◽  
Vol 41 (2) ◽  
pp. 240-244
Author(s):  
Jingbo Xia
Keyword(s):  
Toeplitz Operators ◽  
General Setting ◽  
Unique Ergodicity ◽  
Toeplitz Algebra ◽  
Commutator Ideal ◽  
Uniquely Ergodic

AbstractIt is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial K1-group. We show in this note that if the unique ergodicity is dropped, then such K1-group can be non-trivial. Therefore, in the general setting of minimal flows, even the K-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal.

scholarly journals Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs

2018 ◽  
Vol 40 (5) ◽  
pp. 1351-1401
Author(s):  
MICHEAL PAWLIUK ◽  
MIODRAG SOKIĆ
Keyword(s):  
Automorphism Groups ◽  
Directed Graphs ◽  
Point Of View ◽  
Unique Ergodicity ◽  
Minimal Flow ◽  
Negative Results ◽  
Starting Point ◽  
Complete Multipartite ◽  
Uniquely Ergodic

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the ‘semi-generic’ complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of Angelet al[Random orderings and unique ergodicity of automorphism groups.J. Eur. Math. Soc.,16(2014), 2059–2095], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in Zucker [Amenability and unique ergodicity of automorphism groups of Fraïssé structures.Fund. Math.,226(2014), 41–61]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in Jasińskiet al[Ramsey precompact expansions of homogeneous directed graphs.Electron. J. Combin.,21(4), (2014), 31] and the papers cited therein.

scholarly journals Boshernitzan's criterion for unique ergodicity of an interval exchange transformation

1987 ◽  
Vol 7 (1) ◽  
pp. 149-153 ◽  
Author(s):  
William A. Veech
Keyword(s):  
Unique Ergodicity ◽  
Interval Exchange ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Uniquely Ergodic

AbstractConfirming a conjecture by Boshernitzan, it is proved that ifTis a minimal non-uniquely ergodic interval exchange, the minimum spacing of the partition determined byTnis O(1/n).

scholarly journals Unique ergodicity of the automorphism group of the semigeneric directed graph

2021 ◽  
pp. 1-14
Author(s):  
COLIN JAHEL
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Unique Ergodicity ◽  
Uniquely Ergodic

Abstract We prove that the automorphism group of the semigeneric directed graph (in the sense of Cherlin’s classification) is uniquely ergodic.

A context in which finite or unique ergodicity is generic

2020 ◽  
pp. 1-23
Author(s):  
ANDY Q. YINGST
Keyword(s):  
Unique Ergodicity ◽  
Cantor Space ◽  
Bernoulli Trial ◽  
Residual Set ◽  
Ergodic Measures ◽  
Uniquely Ergodic

Abstract We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

Teichmüller dynamics and unique ergodicity via currents and Hodge theory

2020 ◽  
Vol 2020 (768) ◽  
pp. 39-54
Author(s):  
Curtis T. McMullen
Keyword(s):  
Hodge Theory ◽  
Unique Ergodicity ◽  
Polygonal Billiards ◽  
Teichmuller Dynamics

AbstractWe present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals Pointwise multiple averages for sublinear functions

2018 ◽  
Vol 40 (6) ◽  
pp. 1594-1618
Author(s):  
SEBASTIÁN DONOSO ◽  
ANDREAS KOUTSOGIANNIS ◽  
WENBO SUN
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
Pointwise Convergence ◽  
Limit Function ◽  
Invariant Property ◽  
Ergodic Averages ◽  
Order Zero ◽  
Good For ◽  
Sublinear Functions ◽  
Uniquely Ergodic

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

scholarly journals On a Unique Ergodicity of Some Markov Processes

2011 ◽  
Vol 36 (4) ◽  
pp. 589-606 ◽  
Author(s):  
Rafał Kapica ◽  
Tomasz Szarek ◽  
Maciej Ślȩczka
Keyword(s):  
Markov Processes ◽  
Unique Ergodicity

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

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