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Orbits of homogeneous polynomials on Banach spaces
Ergodic Theory and Dynamical Systems ( IF 0.9 ) Pub Date : 2020-04-13 , DOI: 10.1017/etds.2020.17 RODRIGO CARDECCIA , SANTIAGO MURO
Ergodic Theory and Dynamical Systems ( IF 0.9 ) Pub Date : 2020-04-13 , DOI: 10.1017/etds.2020.17 RODRIGO CARDECCIA , SANTIAGO MURO
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.
中文翻译:
Banach 空间上齐次多项式的轨道
我们研究了 Banach 空间上的齐次多项式引起的动力学。众所周知,在 Banach 空间上定义的齐次多项式不可能有密集的轨道。我们展示了一个齐次多项式的简单而自然的例子,它的轨道同时是$\unicode[STIX]{x1D6FF}$ - 密集(轨道遇到每个半径球$\unicode[STIX]{x1D6FF}$ ),弱稠密,这样$\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ 对每个都是稠密的$\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ 要么是无界的,要么是 0 作为累积点。此外,我们将构造推广到任意无限维可分 Banach 空间。为了证明这一点,我们研究了 Banach 空间上的 Julia 齐次多项式集。
更新日期:2020-04-13
中文翻译:
Banach 空间上齐次多项式的轨道
我们研究了 Banach 空间上的齐次多项式引起的动力学。众所周知,在 Banach 空间上定义的齐次多项式不可能有密集的轨道。我们展示了一个齐次多项式的简单而自然的例子,它的轨道同时是