Capture zones of the family of functions \(\lambda z^m\exp(z)\). (English) Zbl 1052.37038
The article studies the dynamics of the parameter family \(F_{\lambda, m}(z)=\lambda z^m\exp(z)\), more precisely, the components of its Fatou set.
There are two singularities for this family: \(0\) and \(-m\). It is seen that the dynamics of \(z_0=0\) is quite similar for different values of the parameter \(\lambda\). In a computer generated graph, the immediate basin of attraction of \(z_0= 0\) presents an infinite number of strips of the same length that extend to \(+\infty\).
The dynamical plane of the family is influenced by the behavior if iterates at \(z= -m\). Precisely, if \(F_{\lambda,m}(-m)\to\infty\), then the Fatou set of \(F_{\lambda,m}\) is the basin of attraction of \(z_0= 0\). But for those values of \(\lambda\in\mathbb{C}\) for which \(F_{\lambda,m}(-m)\nrightarrow\infty\), it can happen that \(F_{\lambda, m}(-m)\to 0\), that is, the orbit of \(z= -m\) is eventually attracted by \(z_0=0\). The connected sets of such parameters \(\lambda\) are called capture zones of the parametric plane.
Relative to how fast the orbit of \(z= -m\) is captured in the immediate basin of attraction of \(z_0= 0\), the capture zones are classified and analyzed. For instance, it is proved that the \(0\)-zone, also named the main capture zone (i.e., the set of A for which \(z= -m\) belongs to the immediate basin of attraction of \(z_0= 0\)) is bounded and the \(1\)-zone (i.e., \(\{\lambda\mid F_{\lambda,m}(-m)\in\) the immediate basin of attraction of \(z_0=0\)}) is the empty set. The \(2\)-zone is an unbounded region in the half-plane and the \(3\)-zone is made up of infinite horizontal strips.
Reviewers’ remark: As indicated by the present author that a detailed study of the boundary of the capture zone will be appeared in his future paper.
There are two singularities for this family: \(0\) and \(-m\). It is seen that the dynamics of \(z_0=0\) is quite similar for different values of the parameter \(\lambda\). In a computer generated graph, the immediate basin of attraction of \(z_0= 0\) presents an infinite number of strips of the same length that extend to \(+\infty\).
The dynamical plane of the family is influenced by the behavior if iterates at \(z= -m\). Precisely, if \(F_{\lambda,m}(-m)\to\infty\), then the Fatou set of \(F_{\lambda,m}\) is the basin of attraction of \(z_0= 0\). But for those values of \(\lambda\in\mathbb{C}\) for which \(F_{\lambda,m}(-m)\nrightarrow\infty\), it can happen that \(F_{\lambda, m}(-m)\to 0\), that is, the orbit of \(z= -m\) is eventually attracted by \(z_0=0\). The connected sets of such parameters \(\lambda\) are called capture zones of the parametric plane.
Relative to how fast the orbit of \(z= -m\) is captured in the immediate basin of attraction of \(z_0= 0\), the capture zones are classified and analyzed. For instance, it is proved that the \(0\)-zone, also named the main capture zone (i.e., the set of A for which \(z= -m\) belongs to the immediate basin of attraction of \(z_0= 0\)) is bounded and the \(1\)-zone (i.e., \(\{\lambda\mid F_{\lambda,m}(-m)\in\) the immediate basin of attraction of \(z_0=0\)}) is the empty set. The \(2\)-zone is an unbounded region in the half-plane and the \(3\)-zone is made up of infinite horizontal strips.
Reviewers’ remark: As indicated by the present author that a detailed study of the boundary of the capture zone will be appeared in his future paper.
Reviewer: Chung-Chun Yang (Kowloon)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Keywords:
superattracting fixed point; critical point; exponential map; invariant set; basin of attraction; capture zoneReferences:
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