Imperial College London
Alternate

ProfessorSebastianvan Strien

Faculty of Natural Sciences, Department of Mathematics

Chair in Dynamical Systems
 
 
 
//

Contact

 

+44 (0)20 7594 2844s.van-strien Website

 
 
//

Location

 

6M36Huxley BuildingSouth Kensington Campus

//

Summary

 

Publications

Citation

BibTex format

@article{Levin:2020:1361-6544/ab853e,
author = {Levin, G and Shen, W and Strien, SV},
doi = {1361-6544/ab853e},
journal = {Nonlinearity},
pages = {1--43},
title = {Positive Transversality via transfer operators and holomorphic motions  with applications to monotonicity for interval maps},
url = {http://dx.doi.org/10.1088/1361-6544/ab853e},
volume = {33},
year = {2020}
}
Download

RIS format (EndNote, RefMan)

TY  - JOUR
AB  - In this paper we will develop a general approach which shows that generalized"critical relations" of families of locally defined holomorphic maps on thecomplex plane unfold transversally. The main idea is to define a transferoperator, which is a local analogue of the Thurston pullback operator, usingholomorphic motions. Assuming a so-called lifting property is satisfied, weobtain information about the spectrum of this transfer operator and thus abouttransversality. An important new feature of our method is that it is notglobal: the maps we consider are only required to be defined and holomorphic ona neighbourhood of some finite set.  We will illustrate this method by obtaining transversality for a wide classof one-parameter families of interval and circle maps, for example for mapswith flat critical points, but also for maps with complex analytic extensionssuch as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1},for real maps we obtain {\em positive} transversality (where $>0$ holds insteadof 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.
AU  - Levin,G
AU  - Shen,W
AU  - Strien,SV
DO  - 1361-6544/ab853e
EP  - 43
PY  - 2020///
SN  - 0951-7715
SP  - 1
TI  - Positive Transversality via transfer operators and holomorphic motions  with applications to monotonicity for interval maps
T2  - Nonlinearity
UR  - http://dx.doi.org/10.1088/1361-6544/ab853e
UR  - http://arxiv.org/abs/1902.06732v2
UR  - https://iopscience.iop.org/article/10.1088/1361-6544/ab853e/meta
UR  - http://hdl.handle.net/10044/1/79095
VL  - 33
ER  -
Download