Positive Transversality via transfer operators and holomorphic motions
with applications to monotonicity for interval maps
with applications to monotonicity for interval maps
File(s)
Published version
Author(s)
Levin, Genadi
Shen, Weixiao
Strien, Sebastian van
Type
Journal Article
Abstract
In this paper we will develop a general approach which shows that generalized
"critical relations" of families of locally defined holomorphic maps on the
complex plane unfold transversally. The main idea is to define a transfer
operator, which is a local analogue of the Thurston pullback operator, using
holomorphic motions. Assuming a so-called lifting property is satisfied, we
obtain information about the spectrum of this transfer operator and thus about
transversality. An important new feature of our method is that it is not
global: the maps we consider are only required to be defined and holomorphic on
a neighbourhood of some finite set.
We will illustrate this method by obtaining transversality for a wide class
of one-parameter families of interval and circle maps, for example for maps
with flat critical points, but also for maps with complex analytic extensions
such as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1},
for real maps we obtain {\em positive} transversality (where $>0$ holds instead
of just $\ne 0$), and thus monotonicity of entropy for these families, and also
(as an easy application) for the real quadratic family.
This method additionally gives results for unimodal families of the form
$x\mapsto |x|^\ell+c$ for $\ell>1$ not necessarily an even integer and $c$
real.
"critical relations" of families of locally defined holomorphic maps on the
complex plane unfold transversally. The main idea is to define a transfer
operator, which is a local analogue of the Thurston pullback operator, using
holomorphic motions. Assuming a so-called lifting property is satisfied, we
obtain information about the spectrum of this transfer operator and thus about
transversality. An important new feature of our method is that it is not
global: the maps we consider are only required to be defined and holomorphic on
a neighbourhood of some finite set.
We will illustrate this method by obtaining transversality for a wide class
of one-parameter families of interval and circle maps, for example for maps
with flat critical points, but also for maps with complex analytic extensions
such as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1},
for real maps we obtain {\em positive} transversality (where $>0$ holds instead
of just $\ne 0$), and thus monotonicity of entropy for these families, and also
(as an easy application) for the real quadratic family.
This method additionally gives results for unimodal families of the form
$x\mapsto |x|^\ell+c$ for $\ell>1$ not necessarily an even integer and $c$
real.
Date Issued
2020-03-31
Date Acceptance
2020-03-31
Citation
Nonlinearity, 2020, 33 (8), pp.1-43
ISSN
0951-7715
Publisher
IOP Publishing
Start Page
1
End Page
43
Journal / Book Title
Nonlinearity
Volume
33
Issue
8
Copyright Statement
© 2020 IOP Publishing Ltd & London Mathematical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
License URL
Sponsor
Commission of the European Communities
Identifier
http://arxiv.org/abs/1902.06732v2
Grant Number
339523
Subjects
math.DS
math.DS
Publication Status
Published
Date Publish Online
2020-06-10