On the support of the bifurcation measure of cubic polynomials
- PDF / 438,342 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 89 Downloads / 192 Views
Mathematische Annalen
On the support of the bifurcation measure of cubic polynomials Hiroyuki Inou1
· Sabyasachi Mukherjee2
Received: 19 June 2018 / Revised: 15 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by Misiurewicz polynomials. In particular, such parameters admit maximal bifurcations, but do not belong to the support of the bifurcation measure.
Contents 1 Introduction . . . . . . . . . . 2 The slice Per 1 (1) . . . . . . . . 3 Perturbation of parabolic points 4 Proof of Theorem 1 . . . . . . References . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
1 Introduction The connectedness locus C of cubic polynomials is the set of parameters such that the corresponding Julia sets are connected. It is a compact set in the parameter space C2 of all cubic polynomials [3]. For suitable parametrizations, the two critical points of a cubic polynomial can be holomorphically followed throughout the parameter space (see [3,7,9] for various related parametrizations).
Communicated by Ngaiming Mok. Hiroyuki Inou was supported by JSPS KAKENHI Grant Numbers 26400115 and 26287016.
B
Sabyasachi Mukherjee [email protected] Hiroyuki Inou [email protected]
1
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
2
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794, USA
123
H. Inou, S. Mukherjee
In [9], cubic polynomials were parametrized as1 f c,v (z) = z 3 − 3c2 z + 2c3 + v, where c, v ∈ C. The two critical points of f c,v are ±c. The critical point ±c is said to ◦n (±c) be passive near the parameter (c0 , v0 ) if the sequence of functions (c, v) → f c,v 2 forms a normal family in a neighborhood of (c0 , v0 ) in C . Otherwise, ±c is said to be active near (c0 , v0 ). According to [9], the critical point ±c is active precisely on the set ∂C ± , where C ± is the set of parameters for which ±c has bounded orbit. Note that C = C + ∩ C − . The bifurcation locus C bif of cubic polynomials is defined as the complement of the set of all J -stable parameters; i.e. C bif consists of parameters for which at least one critical point is active (see [13, Theorem 4.2] for several equivalent conditions for J -stability). Clearly, we have that C bif = ∂C + ∪ ∂C − ⊃ ∂C. We denote by C ∗ the intersection of the activity loci of the two critical points; i.e. ∗ C := ∂C + ∩ ∂C − ⊂ ∂C. Since C ∗ is the set of parameters for which both critical points are active, it is called the bi-activity locus. DeMarco introduced a natural (1, 1)-current supported exactly on th
Data Loading...
