scholarly journals Lyapunov-maximizing measures for pairs of weighted shift operators

2017 ◽  
Vol 39 (1) ◽  
pp. 225-247
Author(s):  
IAN D. MORRIS
Keyword(s):  
Lyapunov Exponent ◽  
Weighted Shift ◽  
Bounded Operators ◽  
Shift Operators ◽  
Ergodic Measures ◽  
Ergodic Optimization ◽  
Maximizing Measure ◽  
The Matrix ◽  
Two Measures ◽  
Matrix Case

Motivated by recent investigations of ergodic optimization for matrix cocycles, we study the measures of the maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that, for generic pairs of weighted shift operators, the Lyapunov-maximizing measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximizing measure takes any prescribed value less than $\log 2$ for its metric entropy. We also show that, in contrast to the matrix case, the Lyapunov-maximizing measures of pairs of bounded operators are, in general, not characterized by their supports: we construct explicitly a pair of operators and a pair of ergodic measures on the 2-shift with identical supports, such that one of the two measures is Lyapunov-maximizing for the pair of operators and the other measure is not. Our proofs make use of the Ornstein $\overline{d}$-metric to estimate differences in the top Lyapunov exponent of a pair of weighted shift operators as the underlying measure is varied.

CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS

2013 ◽  
Vol 23 (01) ◽  
pp. 1350010 ◽  
Author(s):  
XINXING WU ◽  
PEIYONG ZHU
Keyword(s):  
Positive Integer ◽  
Linear Space ◽  
Normed Linear Space ◽  
Shift Operator ◽  
Weighted Shift ◽  
Constructive Proof ◽  
Sufficient Condition ◽  
Weighted Shift Operator ◽  
Shift Operators ◽  
Devaney Chaos

In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, …) = (μ(0)x1, μ(1)x2, …), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that [Formula: see text] is distributionally chaotic and Li–Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.

Triangular representations of linear algebras

1953 ◽  
Vol 49 (4) ◽  
pp. 595-600 ◽  
Author(s):  
M. P. Drazin
Keyword(s):  
Commutative Algebra ◽  
Similarity Transformation ◽  
Closed Field ◽  
Triangular Form ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Common Eigenvector ◽  
General Systems ◽  
The Matrix ◽  
Matrix Case

It is well known that the elements of any given commutative algebra (and hence of any commutative set) of n × n matrices, over an algebraically closed field K, have a common eigenvector over K; indeed, the elements of such an algebra can be simultaneously reduced to triangular form (by a suitable similarity transformation). McCoy (5) has shown that a triangular reduction is always possible even for matrix algebras satisfying a condition substantially weaker than commutativity. Our aim in this note is to extend these results to more general systems (our arguments being, incidentally, simpler than some used for the matrix case even by writers subsequent to McCoy).

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