scholarly journals Recurrence on affine Grassmannians

2018 ◽  
Vol 39 (12) ◽  
pp. 3207-3223
Author(s):  
YVES BENOIST ◽  
CAROLINE BRUÈRE
Keyword(s):  
Lyapunov Exponent ◽  
Probability Measure ◽  
Affine Group ◽  
Stationary Measure ◽  
Compactly Supported ◽  
Affine Subspaces ◽  
Affine Grassmannians

We study the action of the affine group $G$ of $\mathbb{R}^{d}$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly supported Zariski dense probability measure $\unicode[STIX]{x1D707}$ on $G$, we show that $X_{k,d}$ supports a $\unicode[STIX]{x1D707}$-stationary measure $\unicode[STIX]{x1D708}$ if and only if the $(k+1)\text{th}$ Lyapunov exponent of $\unicode[STIX]{x1D707}$ is strictly negative. In particular, when $\unicode[STIX]{x1D707}$ is symmetric, $\unicode[STIX]{x1D708}$ exists if and only if $2k\geq d$.

Slices of Hewitt–Stromberg measures and co-dimensions formula

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bilel Selmi
Keyword(s):  
Probability Measure ◽  
Borel Probability Measure ◽  
Compactly Supported ◽  
Borel Probability ◽  
The Relationship

Abstract This paper studies the behavior of the lower and upper multifractal Hewitt–Stromberg functions under slices onto ( n - m ) {(n-m)} -dimensional subspaces. More precisely, we discuss the relationship between the multifractal Hewitt–Stromberg functions of a compactly supported Borel probability measure and those of slices or sections of the measure. In addition, we prove that if μ has a finite m-energy and q lies in a certain somewhat restricted interval, then these functions satisfy the expected adding of co-dimensions formula.

scholarly journals A Consistent Markov Partition Process Generated from the Paintbox Process

2011 ◽  
Vol 48 (3) ◽  
pp. 778-791 ◽  
Author(s):  
Harry Crane
Keyword(s):  
Probability Measure ◽  
Point Process ◽  
Markov Processes ◽  
Poisson Point Process ◽  
Transition Probabilities ◽  
Stationary Measure ◽  
Product Measure ◽  
Natural Numbers ◽  
Reversible Processes ◽  
Partition Process

We study a family of Markov processes on P(k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R+ x ∏i=1kP(k) with intensity dt ⊗ ϱν(k), where ϱν is the distribution of the paintbox based on the probability measure ν on Pm, the set of ranked-mass partitions of 1, and ϱν(k) is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

scholarly journals A Consistent Markov Partition Process Generated from the Paintbox Process

2011 ◽  
Vol 48 (03) ◽  
pp. 778-791 ◽  
Author(s):  
Harry Crane
Keyword(s):  
Probability Measure ◽  
Point Process ◽  
Markov Processes ◽  
Poisson Point Process ◽  
Transition Probabilities ◽  
Stationary Measure ◽  
Product Measure ◽  
Natural Numbers ◽  
Reversible Processes ◽  
Partition Process

We study a family of Markov processes onP(k), the space of partitions of the natural numbers with at mostkblocks. The process can be constructed from a Poisson point process onR+x ∏i=1kP(k)with intensity dt⊗ ϱν(k), where ϱνis the distribution of the paintbox based on the probability measure ν onPm, the set of ranked-mass partitions of 1, and ϱν(k)is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

scholarly journals Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

2021 ◽  
pp. 1-40
Author(s):  
ALENA ERCHENKO
Keyword(s):  
Lyapunov Exponent ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Maximal Entropy ◽  
Anosov Diffeomorphism ◽  
Dynamical Invariants ◽  
Area Preserving ◽  
Measure Of Maximal Entropy

Abstract We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

scholarly journals The Furstenberg–Poisson boundary and CAT(0) cube complexes

2017 ◽  
Vol 38 (6) ◽  
pp. 2180-2223 ◽  
Author(s):  
TALIA FERNÓS
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Stationary Measure ◽  
Poisson Boundary ◽  
Proper Action ◽  
Cube Complex ◽  
Finite Dimensional ◽  
Cube Complexes

We show under weak hypotheses that $\unicode[STIX]{x2202}X$, the Roller boundary of a finite-dimensional CAT(0) cube complex $X$ is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group $\unicode[STIX]{x1D6E4}$. In particular, we show that if $\unicode[STIX]{x1D6E4}$ admits a non-elementary proper action on $X$, and $\unicode[STIX]{x1D707}$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\unicode[STIX]{x1D707}$-stationary measure on $\unicode[STIX]{x2202}X$ making it the Furstenberg–Poisson boundary for the $\unicode[STIX]{x1D707}$-random walk on $\unicode[STIX]{x1D6E4}$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

Topological Characteristics of ECG Signal using Lyapunov Exponent and RBF Network

2011 ◽  
Vol 1 (9) ◽  
pp. 53-55
Author(s):  
Abinash Dahal ◽  
◽  
Deepashree Devaraj ◽  
Dr. N. Pradhan Dr. N. Pradhan
Keyword(s):  
Lyapunov Exponent ◽  
Ecg Signal ◽  
Rbf Network ◽  
Topological Characteristics

The complements in an affine group

2013 ◽  
Vol 50 (2) ◽  
pp. 258-265
Author(s):  
Pál Hegedűs
Keyword(s):  
Permutation Group ◽  
Structural Similarity ◽  
Affine Group ◽  
Permutation Module ◽  
Regular Module ◽  
Brauer Characters

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

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