# Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps

## TL;DR

This paper introduces a novel method using transfer operators and holomorphic motions to prove transversality in families of holomorphic maps, leading to results on monotonicity of entropy for interval maps.

## Contribution

It develops a local, holomorphic motion-based approach to establish transversality in parameter families, applicable to maps with flat critical points and polynomial-like extensions.

## Key findings

- Proves transversality for various interval and circle map families.
- Establishes positive transversality and entropy monotonicity for real maps.
- Applies method to unimodal maps with non-integer critical exponents.

## 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.