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Register a new userTransversality for critical relations of families of rational maps: an elementary proof
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In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.
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In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.
In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.
We describe an algorithm for distinguishing hyperbolic components in the parameter space of quadratic rational maps with a periodic critical point. We then illustrate computer images of the hyperbolic components of the parameter spaces V1 - V4, which were produced using our algorithm. We also resolve the singularities of the projective closure of V5 by blowups, giving an alternative proof that as an algebraic curve, the geometric genus of V5 is 1. This explains why we are unable to produce an image for V5.
We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.
An argument is given to associate integrable nonintegrable transition of discrete maps with the transition of Lawveres fixed point theorem to its own contrapositive. We show that the classical description of nonlinear maps is neither complete nor totally predictable.
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