Aubry set for asymptotically sub-additive potentials

2016 ◽  
Vol 16 (02) ◽  
pp. 1660009 ◽  
Author(s):  
Eduardo Garibaldi ◽  
João Tiago Assunção Gomes
Keyword(s):  
Dynamical Systems ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Optimization Theory ◽  
Joint Spectral Radius ◽  
Maximum Value ◽  
Ergodic Optimization ◽  
Aubry Set ◽  
Central Result ◽  
Topological Dynamical Systems

Given a topological dynamical systems [Formula: see text], consider a sequence of continuous potentials [Formula: see text] that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set [Formula: see text] of [Formula: see text]-invariant probabilities that attain the following maximum value [Formula: see text] For this purpose, we extend the notion of Aubry set, denoted by [Formula: see text]. Our central result provides sufficient conditions for the Aubry set to be a maximizing set, i.e. [Formula: see text] belongs to [Formula: see text] if, and only if, its support lies on [Formula: see text]. Furthermore, we apply this result to the study of the joint spectral radius in order to show the existence of periodic matrix configurations approaching this value.

scholarly journals Perfect Orderings on Finite Rank Bratteli Diagrams

2014 ◽  
Vol 66 (1) ◽  
pp. 57-101 ◽  
Author(s):  
S. Bezuglyi ◽  
J. Kwiatkowski ◽  
R. Yassawi
Keyword(s):  
Dynamical Systems ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
The Other ◽  
Bratteli Diagram ◽  
Incidence Matrices ◽  
Bratteli Diagrams ◽  
Topological Dynamical Systems ◽  
Almost All ◽  
Necessary And Sufficient

AbstractGiven a Bratteli diagram B, we study the set 𝒪B of all possible orderings on B and its subset PB consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering ω to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram B. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram B of rank k, we endow the set 𝒪B with product measure μ and prove that there is some 1 ≤ j ≤ k such that μ-almost all orderings on B have j maximal and j minimal paths. If j is strictly greater than the number of minimal components that B has, then μ-almost all orderings are imperfect.

REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Fractals ◽  
2010 ◽  
Vol 18 (04) ◽  
pp. 483-498 ◽  
Author(s):  
IBRAHIM KIRAT ◽  
ILKER KOCYIGIT
Keyword(s):  
Convex Hull ◽  
Lebesgue Measure ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Convex Hulls ◽  
Compact Set ◽  
Joint Spectral Radius ◽  
Digit Set ◽  
Necessary And Sufficient ◽  
Digit Expansions

Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

scholarly journals Ruelle operator theorem for non-expansive systems

2009 ◽  
Vol 30 (2) ◽  
pp. 469-487 ◽  
Author(s):  
YUNPING JIANG ◽  
YUAN-LING YE
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Iterated Function Systems ◽  
Ruelle Operator ◽  
Iterated Function

AbstractThe Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

On the Almost Sure Stability of Linear Stochastic Distributed-Parameter Dynamical Systems

1966 ◽  
Vol 33 (1) ◽  
pp. 182-186 ◽  
Author(s):  
P. K. C. Wang
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Integral Equations ◽  
Sufficient Conditions ◽  
Distributed Parameter ◽  
Almost Sure Stability ◽  
Partial Differential ◽  
Stochastic Parameters

In this paper, sufficient conditions for almost sure stability and asymptotic stability of certain classes of linear stochastic distributed-parameter dynamical systems are derived. These systems are described by a set of linear partial differential or differential-integral equations with stochastic parameters. Various examples are given to illustrate the application of the main results.

Ergodic optimization for noncompact dynamical systems

2007 ◽  
Vol 22 (3) ◽  
pp. 379-388 ◽  
Author(s):  
O. Jenkinson ◽  
R. D. Mauldin ◽  
M. Urbański
Keyword(s):  
Dynamical Systems ◽  
Ergodic Optimization
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