Refined scales of weak-mixing dynamical systems: typical behaviour

2019 ◽  
Vol 40 (12) ◽  
pp. 3296-3309
Author(s):  
SILAS L. CARVALHO ◽  
CÉSAR R. DE OLIVEIRA
Keyword(s):  
Invariant Measures ◽  
Dynamical Behaviour ◽  
Lebesgue Spaces ◽  
Weak Mixing ◽  
Axiom A ◽  
Typical Behaviour ◽  
Topologically Mixing ◽  
Continuous Measures ◽  
Basic Sets ◽  
Measure Preserving Transformations

We study sets of measure-preserving transformations on Lebesgue spaces with continuous measures taking into account extreme scales of variations of weak mixing. It is shown that the generic dynamical behaviour depends on subsequences of time going to infinity. We also present corresponding generic sets of (probability) invariant measures with respect to topological shifts over finite alphabets and Axiom A diffeomorphisms over topologically mixing basic sets.

Postcritical sets and saddle basic sets for Axiom A polynomial skew products on

2012 ◽  
Vol 33 (4) ◽  
pp. 1124-1145 ◽  
Author(s):  
SHIZUO NAKANE
Keyword(s):  
Axiom A ◽  
Skew Products ◽  
Basic Sets

AbstractWe investigate the link between postcritical behaviors and the relations of saddle basic sets for Axiom A polynomial skew products on $\mathbb {C}^2$, and characterize various properties concerning the three kinds of accumulation sets defined by DeMarco and Hruska [Axiom A polynomial skew products of $\mathbb {C}^2$ and their postcritical sets. Ergod. Th. & Dynam. Sys.28(2008), 1749–1779] in terms of the saddle basic sets. We also give a new example of higher degree.

The ambient structure of basic sets

1995 ◽  
Vol 15 (6) ◽  
pp. 1183-1188
Author(s):  
A. Löffler
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Axiom A ◽  
Basic Set ◽  
Local Product ◽  
Basic Sets

AbstractLet Λ be a basic set of an Axiom A diffeomorphism of a compact Riemannian manifold M without boundary. If ε is small enough one can find by local product structure that for x ε Λ there is a neighborhood V(x) in M such that V ∩ Λ is homeomorphic to . The author proves that this homeomorphism can be extended to a homeomorphism of V onto .

INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS

2009 ◽  
Vol 19 (01) ◽  
pp. 409-417 ◽  
Author(s):  
MD SHAFIQUL ISLAM
Keyword(s):  
Invariant Measures ◽  
Stochastic Matrix ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Higher Dimension ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Higher Dimensional ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

scholarly journals The Chebotarov theorem for Galois coverings of Axiom A flows

1986 ◽  
Vol 6 (1) ◽  
pp. 133-148 ◽  
Author(s):  
William Parry ◽  
Mark Pollicott
Keyword(s):  
Asymptotic Formula ◽  
Conjugacy Class ◽  
Compact Group ◽  
Negative Curvature ◽  
Basic Method ◽  
Frame Bundle ◽  
Axiom A ◽  
Closed Orbits ◽  
Galois Coverings ◽  
Basic Sets

AbstractWe consider G (Galois) coverings of Axiom A flows (restricted to basic sets) and prove an analogue of Chebotarev's theorem. The theorem provides an asymptotic formula for the number of closed orbits whose Frobenius class is a given conjugacy class in G. An application answers a question raised by J. Plante. The basic method is then extended to compact group extensions and applied to frame bundle flows defined on manifolds of variable negative curvature.

scholarly journals Correction to ‘Equivariant spectral decomposition for flows with a ℤ-action’

1990 ◽  
Vol 10 (4) ◽  
pp. 787-791 ◽  
Author(s):  
Lee Mosher
Keyword(s):  
Spectral Decomposition ◽  
Decomposition Theorem ◽  
Anosov Flow ◽  
Axiom A ◽  
Basic Sets

The statement and proof of the ℤ-spectral decomposition theorem for a pseudo-Anosov flow φ on a 3-manifold M are in error (see [1]). There are counter-examples which show that the theorem as stated is false. We remark that the results of § 9 of [1], concerning analogues of the ℤ-spectral decomposition theorem for basic sets of Axiom A flows, are unaffected by this error.

A Bound on the Number of Invariant Measures

1981 ◽  
Vol 24 (1) ◽  
pp. 123-124 ◽  
Author(s):  
Abraham Boyarsky
Keyword(s):  
Upper Bound ◽  
Invariant Measures ◽  
Absolutely Continuous ◽  
Continuous Measures ◽  
Absolutely Continuous Measures ◽  
Points Of Discontinuity

For τ a piecewise C2 transformation, we present a method for obtaining an upper bound for the number of independent absolutely continuous measures invariant under τ.Let τ = [0,1] and let τ:I→ J be a piecewise C2 transformation with infI1 |dτ/dx| > 1, where I1 = I-P and P denotes the points of discontinuity of τ and τ′

On a class of stable conditional measures

2010 ◽  
Vol 31 (5) ◽  
pp. 1499-1515 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Invariant Measures ◽  
Reversible Systems ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Equilibrium Measures ◽  
Basic Set ◽  
Stable Dimension ◽  
Saddle Type ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractThe dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

scholarly journals Ergodic and mixing sequences of transformations

1984 ◽  
Vol 4 (3) ◽  
pp. 353-366 ◽  
Author(s):  
Daniel Berend ◽  
Vitaly Bergelson
Keyword(s):  
Probability Space ◽  
Strong Mixing ◽  
Compact Groups ◽  
Ergodic Theorems ◽  
Affine Transformations ◽  
Weak Mixing ◽  
Mixing Sequences ◽  
Measure Preserving Transformations

AbstractThe notions of ergodicity, strong mixing and weak mixing are defined and studied for arbitrary sequences of measure-preserving transformations of a probability space. Several results, notably ones connected with mean ergodic theorems, are generalized from the case of the sequence of all powers of a single transformation to this case. The conditions for ergodicity, strong mixing and weak mixing of sequences of affine transformations of compact groups are investigated.

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