Leading monomials of escape regions

This chapter considers a leading monomial vector, which uniquely determines the escape region and within which is encoded important information about the limiting behavior of the periodic critical orbit. Thus, for each p ≥ 1, this chapter considers the affine algebraic curve Sₚ formed by all monic and centered cubic polynomials with a marked critical point which has period p under iterations. Each unbounded hyperbolic component of Sₚ, called an escape region, has an associated vector of leading monomials which encodes the asymptotic behavior of the periodic critical orbit. The chapter shows that this vector determines the escape region, giving a positive answer to a question posed by Bonifant and Milnor.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

A point is normal for almost all maps βx+α mod 1 or generalized β-transformations

We consider the map Tα,β(x):=βx+α mod 1, which admits a unique probability measure μα,β of maximal entropy. For x∈[0,1], we show that the orbit of x is μα,β-normal for almost all (α,β)∈[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit of x=0 is μα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalized β-transformations (in particular, the tent map). We show that the critical orbit x=1 is normal with respect to the measure of maximal entropy for almost all β.