scholarly journals Effectivity of uniqueness of the maximal entropy measure on -adic homogeneous spaces

2015 ◽  
Vol 36 (6) ◽  
pp. 1972-1988 ◽  
Author(s):  
RENE RÜHR
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Algebraic Group ◽  
Homogeneous Spaces ◽  
Regular Representation ◽  
Invariant Probability Measure ◽  
Linear Algebraic Group ◽  
Maximal Entropy ◽  
Entropy Measure ◽  
Effective Version

We consider the dynamical system given by an $\text{Ad}$-diagonalizable element $a$ of the $\mathbb{Q}_{p}$-points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$. Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$-finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$-invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$, then ${\it\mu}$ is close to the unique $G$-invariant probability measure of $X$.

scholarly journals Statistical properties of the maximal entropy measure for partially hyperbolic attractors

2016 ◽  
Vol 37 (4) ◽  
pp. 1060-1101 ◽  
Author(s):  
ARMANDO CASTRO ◽  
TEÓFILO NASCIMENTO
Keyword(s):  
Probability Measure ◽  
Hyperbolic Systems ◽  
Maximal Entropy ◽  
Entropy Measure ◽  
Hölder Continuous ◽  
Hyperbolic Diffeomorphisms ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

Algorithmic complexity of points in dynamical systems

1993 ◽  
Vol 13 (4) ◽  
pp. 807-830 ◽  
Author(s):  
Homer S. White
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Invariant Probability Measure ◽  
Algorithmic Complexity ◽  
Maximal Entropy ◽  
Property A ◽  
Open Set ◽  
Unique Measure ◽  
Set Of Points

AbstractThis work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

scholarly journals A note on the mean value theorem for special homogeneous spaces

1996 ◽  
Vol 143 ◽  
pp. 111-117 ◽  
Author(s):  
Masanori Morishita ◽  
Takao Watanabe
Keyword(s):  
Algebraic Group ◽  
Algebraic Variety ◽  
Homogeneous Spaces ◽  
Rational Numbers ◽  
Rational Points ◽  
Linear Algebraic Group ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean

Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

scholarly journals Rigidity for group actions on homogeneous spaces by affine transformations

2016 ◽  
Vol 37 (7) ◽  
pp. 2060-2076
Author(s):  
MOHAMED BOULJIHAD
Keyword(s):  
Probability Measure ◽  
Lie Groups ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Betti Numbers ◽  
Invariant Probability Measure ◽  
Equivalence Relations ◽  
Nilpotent Lie Groups ◽  
Affine Transformations ◽  
Rigidity Property

We give a criterion for the rigidity of the action of a group of affine transformations of a homogeneous space of a real Lie group. Let $G$ be a real Lie group, $\unicode[STIX]{x1D6EC}$ a lattice in $G$, and $\unicode[STIX]{x1D6E4}$ a subgroup of the affine group $\text{Aff}(G)$ stabilizing $\unicode[STIX]{x1D6EC}$. Then the action of $\unicode[STIX]{x1D6E4}$ on $G/\unicode[STIX]{x1D6EC}$ has the rigidity property in the sense of Popa [On a class of type $\text{II}_{1}$ factors with Betti numbers invariants. Ann. of Math. (2)163(3) (2006), 809–899] if and only if the induced action of $\unicode[STIX]{x1D6E4}$ on $\mathbb{P}(\mathfrak{g})$ admits no $\unicode[STIX]{x1D6E4}$-invariant probability measure, where $\mathfrak{g}$ is the Lie algebra of $G$. This generalizes results of Burger [Kazhdan constants for $\text{SL}(3,\mathbf{Z})$. J. Reine Angew. Math.413 (1991), 36–67] and Ioana and Shalom [Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn.7(2) (2013), 403–417]. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to two-step nilpotent Lie groups.

scholarly journals The structure of tame minimal dynamical systems

2007 ◽  
Vol 27 (6) ◽  
pp. 1819-1837 ◽  
Author(s):  
ELI GLASNER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Probability Measure ◽  
Topological Space ◽  
Haar Measure ◽  
Invariant Probability Measure ◽  
Structure Theory ◽  
Convergence Of Sequences ◽  
Minimal Dynamical Systems ◽  
Minimal Dynamical System

AbstractA dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

scholarly journals Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

2021 ◽  
Vol 157 (12) ◽  
pp. 2657-2698
Author(s):  
Runlin Zhang
Keyword(s):  
Invariant Measure ◽  
Algebraic Group ◽  
Present Article ◽  
Homogeneous Spaces ◽  
Natural Extension ◽  
Complete Classification ◽  
Linear Algebraic Group ◽  
Arbitrary Sequence ◽  
Limiting Behavior ◽  
Arithmetic Lattice

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$ , let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$ -subgroup. There is a $H$ -invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$ . Given a sequence $(g_n)$ in $G$ , we study the limiting behavior of $(g_n)_*\mu _H$ under the weak- $*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$ . We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

A minimal distal map on the torus with sub-exponential measure complexity

2018 ◽  
Vol 40 (4) ◽  
pp. 953-974 ◽  
Author(s):  
WEN HUANG ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Borel Probability Measure ◽  
Skew Product ◽  
Topological Dynamical System ◽  
Borel Probability ◽  
Product Map

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

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