Shorter Notes: Erratum to "Infinite-Dimensional Jacobi Matrices Associated with Julia Sets"

1984 ◽  
Vol 92 (1) ◽  
pp. 156
Author(s):  
M. F. Barnsley ◽  
J. S. Geronimo ◽  
A. N. Harrington
Keyword(s):  
Julia Sets ◽  
Jacobi Matrices ◽  
Infinite Dimensional

scholarly journals Infinite-dimensional Jacobi matrices associated with Julia sets

1983 ◽  
Vol 88 (4) ◽  
pp. 625-625 ◽  
Author(s):  
M. F. Barnsley ◽  
J. S. Geronimo ◽  
A. N. Harrington
Keyword(s):  
Julia Sets ◽  
Jacobi Matrices ◽  
Infinite Dimensional

scholarly journals Almost periodic Jacobi matrices associated with Julia sets for polynomials

1985 ◽  
Vol 99 (3) ◽  
pp. 303-317 ◽  
Author(s):  
M. F. Barnsley ◽  
J. S. Geronimo ◽  
A. N. Harrington
Keyword(s):  
Julia Sets ◽  
Jacobi Matrices ◽  
Almost Periodic

scholarly journals Orbits of homogeneous polynomials on Banach spaces

2020 ◽  
pp. 1-29
Author(s):  
RODRIGO CARDECCIA ◽  
SANTIAGO MURO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Homogeneous Polynomial ◽  
Julia Sets ◽  
Accumulation Point ◽  
Dense Orbit ◽  
Homogeneous Polynomials ◽  
Infinite Dimensional

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$ -dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$ ), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

scholarly journals A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
S. B. Rutkevich
Keyword(s):  
Spectral Properties ◽  
Jacobi Operator ◽  
Scattering Data ◽  
Jacobi Matrices ◽  
Reflection Symmetry ◽  
Matrix Elements ◽  
Discrete Schrödinger Operator ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Secondary Diagonal

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.

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