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

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 (Tsujii M 1994 A note on Milnor and Thurston’s monotonicity theorem Geometry and Analysis in Dynamical System vol 14 (Singapore: World Scientific) pp 60–2; Tsujii M 2000 Ergod. Theor. Dyn. Syst. 20 925–933), for real maps we obtain positive transversality (where >0 holds instead of just ≠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 ↦ |x| ℓ + c for ℓ > 1 not necessarily an even integer and c real.

Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

“…with α, β > 1 large, then there are partial results towards monotonicity in [31] see also [32]. Monotonicity for this family is only known in full generality when α = β is an even integer.…”

“…Monotonicity for this family is only known in full generality when α = β is an even integer. For references on the history of results on monotonicity, see [32].…”

“…Since β > 1 we see that if k is large enough, we get huge improvement on the relative size of extension interval [A k+2 , B k+2 ] compared to the renormalization interval [a k+2 , b k+2 ]. From this point the argument can be applied inductively and(32) follows. Lemma 8 gives |a k+1 − b k+1 | ≈ |a k − b k |.…”

Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

“…3 for the logistic family, which is actually the same after change of coordinates). In recent paper [11], the monotonicity result was proven for the family x + c with large (not necessarily integer). Our family of maps is different from those families in the sense that we allow infinite derivatives at the discontinuity point, which makes the problem even more complicated because the complex analysis technique doesn't work here.…”

Fixed points of the Ruelle–Thurston operator and the Cauchy transform

“…We give an example of application after Theorem 1 of this paper. For an alternative (local) approach and discussions, see [20].…”