共查询到20条相似文献,搜索用时 31 毫秒
1.
L. J. Díaz K. Gelfert M. Rams 《Proceedings of the Steklov Institute of Mathematics》2017,297(1):98-115
We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents. 相似文献
2.
David Burguet 《Inventiones Mathematicae》2011,186(1):191-236
We prove that C2\mathcal{C}^{2} surface diffeomorphisms have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet.
Following the strategy of Downarowicz and Maass (Invent. Math. 176:617–636, 2009) we bound the local entropy of ergodic measures in terms of Lyapunov exponents. This is done by reparametrizing Bowen balls
by contracting maps in a approach combining hyperbolic theory and Yomdin’s theory. 相似文献
3.
We study the ergodic theory of non-conservative C
1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose
supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C
1-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem
by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C
1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ. 相似文献
4.
《中国科学 数学(英文版)》2020,(9)
We study various weaker forms of the inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called ergodic inverse shadowing property(Birkhoff averages of continuous functions along an exact trajectory and the approximating one are close). We demonstrate that this property implies the continuity of the set of invariant measures in the Hausdorff metric. We show that the class of systems with ergodic inverse shadowing is quite broad; it includes all diffeomorphisms with hyperbolic nonwandering sets. Second, we study the so-called individual inverse shadowing(any exact trajectory can be traced by approximate ones, but this shadowing is not uniform with respect to the initial point of the trajectory).We demonstrate that this property is closely related to structural stability and ?-stability of diffeomorphisms. 相似文献
5.
We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are
given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents
the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is
based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on
spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such
groups. Finally, we examine the structure of self-joinings of extensions.
We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively
a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss
on the class of automorphisms disjoint from all weakly mixing transformations.
Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker
dynamical systems.
Research partially supported by KBN grant 2 P03A 002 14 (1998). 相似文献
6.
Jon Aaronson Hitoshi Nakada Omri Sarig Rita Solomyak 《Israel Journal of Mathematics》2002,128(1):93-134
For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures.
These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic
ergodic theorems, and bounded rational ergodicity. 相似文献
7.
Pei-Dong Liu Jian-Sheng Xie 《Transactions of the American Mathematical Society》2006,358(9):3751-3780
We consider dynamics of compositions of stationary random diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.
8.
Robert J. Zimmer 《Israel Journal of Mathematics》1980,35(4):289-300
We consider three problems concerning cocycles of ergodic group actions: behavior of cohomology under the restriction of an
ergodic semi-simple Lie group action to a lattice subgroup; “compactness” of bounded cocyles; and the relation of relative
almost periodicity to relative discrete spectrum for extensions of ergodic actions. 相似文献
9.
We consider smooth (not necessarily invertible) maps of Hilbert spaces preserving ergodic Borel probability measures, and
prove the existence of hyperbolic periodic orbits and horseshoes in the absence of zero Lyapunov exponents. These results
extend Katok’s work on diffeomorphisms of compact manifolds to infinite dimensions, with potential applications to some classes
of periodically forced PDEs. 相似文献
10.
R. C. Fabec 《Israel Journal of Mathematics》1981,40(2):175-186
We demonstrate that normal ergodic extensions of group actions are characterized as skew product actions given by cocycles
into locally compact groups. As a consequence, Robert Zimmer’s characterization of normal ergodic group actions is generalized
to the noninvariant case. We also obtain the uniqueness theorem which generalizes the von Neumann Halmos uniqueness theorem
and Zimmer’s uniqueness theorem for normal actions with relative discrete spectrum. 相似文献
11.
Leonid Polterovich 《Journal of the European Mathematical Society》1999,1(1):87-107
Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S1→G is called strictly ergodic if for some irrational number α the associated skew product map T:S1×Y→S1×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements
of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic
loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on
the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove
that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology.
Received July 7, 1998 / final version received September 14, 1998 相似文献
12.
In this note we show that all diffeomorphisms close enough to the time-one map of the frame flow on certain negatively curved manifolds are ergodic. As a simple corollary we deduce that the frame flows are ergodic for all compact manifolds with curvature pinched sufficiently close to –1, thus providing results in the case of manifolds of dimension 7 or 8 which were missing from the results of Brin and Karcher. 相似文献
13.
In the space of diffeomorphisms of an arbitrary closed manifold of dimension ≥ 3, we construct an open set such that each difteomorphism in this set has an invariant ergodic measure with respect to which one of its Lyapunov exponents is zero. These difteomorphisins are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with the circle as the fiber. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products. 相似文献
14.
We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity. We first deal with them in the case of conservative homeomorphisms of a manifold and C
1-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenbergs celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]). 相似文献
15.
We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé. 相似文献
16.
Towards a Lie theory of locally convex groups 总被引:3,自引:0,他引:3
Karl-Hermann Neeb 《Japanese Journal of Mathematics》2006,1(2):291-468
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled
on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability
of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group
can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization
to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of
diffeomorphisms. 相似文献
17.
Qing Zhang 《Israel Journal of Mathematics》1995,92(1-3):249-261
We give an example to show that compact extensions of Kronecker factors for two ergodic commuting measure preserving transformations
can be different. Also, a criteria for ergodicity of skew products of irrational rotations with finite abelian groups is obtained. 相似文献
18.
In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example,
all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably
ergodic. Our main tools are the new concepts – julienne density point and julienne quasi-conformality of the stable and unstable
holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.
Received June 14, 1999 / final version received October 25, 1999 相似文献
19.
We show that a broad class of extensions of measure preserving systems in the context of ergodic theory can be realized by
topological models for which the extension is “almost one-one”. 相似文献
20.
A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. 相似文献