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.

Some counterexamples in topological dynamics

2008 ◽  
Vol 28 (4) ◽  
pp. 1291-1322 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Topological Dynamics ◽  
Residual Set ◽  
Topologically Mixing ◽  
Uniquely Ergodic

AbstractIn this paper, we exhibit, for any sparse-enough increasing sequence {pn} of integers, totally minimal, totally uniquely ergodic, and topologically mixing systems (X,T) and (X′,T′) and f∈C(X) for which the averages $({1}/{N}) \sum _{n=0}^{N-1} f(T^{p_n} x)$ fail to converge on a residual set in X, and where there exists x′∈X′ with $x' \notin \overline {\{T'^{p_n} x'\}}$.

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 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

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.

scholarly journals Graph covers and ergodicity for zero-dimensional systems

2014 ◽  
Vol 36 (2) ◽  
pp. 608-631 ◽  
Author(s):  
TAKASHI SHIMOMURA
Keyword(s):  
Linear Dependence ◽  
Invariant Measures ◽  
Unique Ergodicity ◽  
Dimensional System ◽  
General Graph ◽  
Bratteli Diagrams ◽  
Linearly Independent ◽  
Ergodic Measures ◽  
Time Average ◽  
Path Number

Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.

scholarly journals DIMENSION THEORY OF LINEAR SOLENOIDS

Fractals ◽  
2007 ◽  
Vol 15 (01) ◽  
pp. 63-72 ◽  
Author(s):  
JÖRG NEUNHÄUSERER
Keyword(s):  
Dimension Theory ◽  
Ergodic Measures ◽  
Full Dimension ◽  
Fractal Attractor

We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question of whether there exists a measure of full dimension.

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.

Sign in / Sign up

Export Citation Format

Share Document