Orbits of homogeneous polynomials on Banach spaces
Keyword(s):
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.
2006 ◽
Vol 49
(1)
◽
pp. 39-52
◽
Keyword(s):
2019 ◽
Vol 38
(3)
◽
pp. 133-140
Keyword(s):
1996 ◽
Vol 1
(4)
◽
pp. 381-396
◽
Keyword(s):
2011 ◽
Vol 53
(3)
◽
pp. 443-449
◽
2005 ◽
Vol 2005
(24)
◽
pp. 3895-3908
◽
2007 ◽
Vol 50
(4)
◽
pp. 619-631
◽