BOUNDED ORBITS AND BOUNDED SHADOWING FOR LINEAR DYNAMICS
Abstract
We study the relation between shadowing, bounded orbits, and hyperbolicity for bounded linear operators on complex Banach spaces. Our purpose is two-fold. First, we prove that a not necessarily invertible operator is hyperbolic if and only if it has the positive shadowing property and its subspace of forward-bounded orbits is closed. This complements the characterization of hyperbolicity obtained by Lee and Morales for invertible operators. Second, we give a complete spectral characterization of the bounded shadowing property. Writing M = E c (T ), where E c (T ) consists of the vectors with bounded two-sided orbits, we show that bounded shadowing is equivalent to shadowing of T | M and uniform expansivity of the operator induced by T on X/M . Equivalently, the surjectivity spectrum of the restriction and the approximate point spectrum of the quotient operator avoid the unit circle. The proofs rely on the spectral characterization of shadowing established by Dragičević and Pituk. We derive several consequences and give examples distinguishing shadowing, bounded shadowing, and density of bounded orbits.
Related
- Shadowing and Topological Stability via Generalized Pseudo-Hyperbolicity in Autonomous and Nonautonomous Linear Dynamics
- Bounded orbits on homogeneous spaces and bounded geodesics on locally symmetric spaces
- Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits
- Linear Stability of Double-Double Orbits in the Equal-Mass Four-Body Problem via Index Theory
- Linear Independence of Polynomial Compositions with Bounded Input Degree
BibTeX
@misc{dragicevic2026bounded,
title = {{BOUNDED ORBITS AND BOUNDED SHADOWING FOR LINEAR DYNAMICS}},
author = {Davor Dragičević},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05838111v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - BOUNDED ORBITS AND BOUNDED SHADOWING FOR LINEAR DYNAMICS
AU - Davor Dragičević
PY - 2026
DA - 2026/10/24
PB - HAL
UR - https://hal.science/hal-05838111v1
AB - We study the relation between shadowing, bounded orbits, and hyperbolicity for bounded linear operators on complex Banach spaces. Our purpose is two-fold. First, we prove that a not necessarily invertible operator is hyperbolic if and only if it has the positive shadowing property and its subspace of forward-bounded orbits is closed. This complements the characterization of hyperbolicity obtained by Lee and Morales for invertible operators. Second, we give a complete spectral characterization of the bounded shadowing property. Writing M = E c (T ), where E c (T ) consists of the vectors with bounded two-sided orbits, we show that bounded shadowing is equivalent to shadowing of T | M and uniform expansivity of the operator induced by T on X/M . Equivalently, the surjectivity spectrum of the restriction and the approximate point spectrum of the quotient operator avoid the unit circle. The proofs rely on the spectral characterization of shadowing established by Dragičević and Pituk. We derive several consequences and give examples distinguishing shadowing, bounded shadowing, and density of bounded orbits.
ER -