Bifurcations of periodic points of holomorphic maps from \(\mathbb{C}^2\) into \(\mathbb{C}^2\).

*(English)*Zbl 0974.32011Let \(F: \mathbb{C}^n\to \mathbb{C}^n\) be a holomorphic map, \(F^k\) be the \(k\)-th iteration of \(F\) and let \(p\in\mathbb{C}^n\) be a periodic point of \(F\) of period \(k\). That is to say, \(F^k(p)= p\) but for any positive integer \(j\) with \(j<k\), \(F^j(p)\neq p\). If \(p\) is hyperbolic, namely \(DF^k(p)\) has no eigenvalue of modulus 1, then it is well known that the dynamical behavior of \(F\) is stable near the periodic orbit \(\Gamma= \{p, F(p),\dots, F^{k-1}(p)\}\). But if \(\Gamma\) is not hyperbolic, the dynamical behavior of \(F\) near \(\Gamma\) may be very complicated and unstable. In this case, a very interesting bifurcational phenomenon may occur even though \(\Gamma\) may be the only periodic orbit in some neighborhood of \(\Gamma\): for given \(M\in \mathbb{N}\setminus \{1\}\) there may exist a \(C^r\)-arc \(F_t\) in the space \(\mathbb{H}(\mathbb{C}^n)\) of holomorphic maps from \(\mathbb{C}^n\) into \(\mathbb{C}^n\) \((r\in\mathbb{N}\) or \(r=\infty)\), \(t\in [0,1]\), such that \(F_0= F\) and for \(t\in (0,1]\), \(F_t\) has an \(Mk\)-periodic orbit \(\Gamma_t\) with \(d(\Gamma_t,\Gamma)= \sup_{p\in\Gamma_t} \inf_{q\in\Gamma} \|p-q\|\to 0\) as \(t\to 0\). The period increases in the way by a factor \(M\) under a \(C^r\)-small perturbation! If such \(F_t\) does exist, then \(\Gamma\), as well as \(p\), is said to be \(M\)-tupling bifurcational. This definition is independent of \(r\).

For the above \(F\) there may exist a \(C^r\)-arc \(F_t^*\) in \(\mathbb{H}(\mathbb{C}^n)\), \(t\in [0,1]\), such that \(F_0^*= F\) and for \(t\in (0,1]\), \(F_t^*\) has two distinct \(k\)-periodic orbits \(\Gamma_{t,1}\) and \(\Gamma_{t,2}\) with \(d(\Gamma_{t,i}, \Gamma)\to 0\) as \(t\to 0\), \(i=1,2\). If such \(F_t^*\) does exist, \(\Gamma\), as well as \(p\), is said to be 1-tupling bifurcational.

In recent decades, there have been many papers and remarkable results which deal with period doubling bifurcations of periodic orbits of parametrized maps. L. Block and D. Hart pointed out that period \(M\)-tupling bifurcations can not occur for \(M>2\) in one-dimensional case. There are examples showing that for any \(M\in\mathbb{N}\), period \(M\)-tupling bifurcations can occur for certain parametrized maps in higher-dimensional cases. An \(M\)-tupling bifurcational periodic orbit defined here acts as a critical orbit which leads to period \(M\)-tupling bifurcations in some parametrized maps. The main result of this paper is:

Theorem. Let \(k\in\mathbb{N}\), \(M\in\mathbb{N}\) and let \(F: \mathbb{C}^2\to \mathbb{C}^2\) be a holomorphic map with \(k\)-periodic point \(p\). Then \(p\) is \(M\)-tupling bifurcational if and only if \(DF^k(p)\) has a nonzero periodic point of period \(M\).

For the above \(F\) there may exist a \(C^r\)-arc \(F_t^*\) in \(\mathbb{H}(\mathbb{C}^n)\), \(t\in [0,1]\), such that \(F_0^*= F\) and for \(t\in (0,1]\), \(F_t^*\) has two distinct \(k\)-periodic orbits \(\Gamma_{t,1}\) and \(\Gamma_{t,2}\) with \(d(\Gamma_{t,i}, \Gamma)\to 0\) as \(t\to 0\), \(i=1,2\). If such \(F_t^*\) does exist, \(\Gamma\), as well as \(p\), is said to be 1-tupling bifurcational.

In recent decades, there have been many papers and remarkable results which deal with period doubling bifurcations of periodic orbits of parametrized maps. L. Block and D. Hart pointed out that period \(M\)-tupling bifurcations can not occur for \(M>2\) in one-dimensional case. There are examples showing that for any \(M\in\mathbb{N}\), period \(M\)-tupling bifurcations can occur for certain parametrized maps in higher-dimensional cases. An \(M\)-tupling bifurcational periodic orbit defined here acts as a critical orbit which leads to period \(M\)-tupling bifurcations in some parametrized maps. The main result of this paper is:

Theorem. Let \(k\in\mathbb{N}\), \(M\in\mathbb{N}\) and let \(F: \mathbb{C}^2\to \mathbb{C}^2\) be a holomorphic map with \(k\)-periodic point \(p\). Then \(p\) is \(M\)-tupling bifurcational if and only if \(DF^k(p)\) has a nonzero periodic point of period \(M\).

Reviewer: Guang Yuan Zhang (Beijing)