Mating quadratic maps with the modular group II

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences $$\mathcal {F}_a$$Fa: $$\begin{aligned} \left( \frac{aw-1}{w-1}\right) ^2+\left( \frac{aw-1}{w-1}\right) \left( \frac{az+1}{z+1}\right) +\left( \frac{az+1}{z+1}\right) ^2=3 \end{aligned}$$aw-1w-12+aw-1w-1az+1z+1+az+1z+12=3and proved that for every value of $$a \in [4,7] \subset \mathbb {R}$$a∈[4,7]⊂R the correspondence $$\mathcal {F}_a$$Fa is a mating between a quadratic polynomial $$Q_c(z)=z^2+c,\,\,c \in \mathbb {R}$$Qc(z)=z2+c,c∈R, and the modular group $$\varGamma =PSL(2,\mathbb {Z})$$Γ=PSL(2,Z). They conjectured that this is the case for every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus. We show here that matings between the modular group and rational maps in the parabolic quadratic family $$Per_1(1)$$Per1(1) provide a better model: we prove that every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus is such a mating.

On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

Let$\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$and$T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$. Let$A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$be the unique compact set satisfying$A=T_{1}(A)\cup T_{-1}(A)$. In this paper, we give a detailed analysis of$A$and the parameters$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$where$A$satisfies various topological properties. In particular, we show that if$\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$, then$A$has a non-empty interior, thus significantly improving the bound from Dajaniet al[Self-affine sets with positive Lebesgue measure.Indag. Math. (N.S.)25(2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series.Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of$A$which have a unique address has positive Hausdorff dimension for all$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$. Finally, we investigate simultaneous$(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$-expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings.Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference2008, pp. 743–748].

Multicorns are not path connected

This chapter proves that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z ↦ ̄z² + c. The chapter extends this discussion more generally for antiholomorphic unicritical polynomials of degrees d ≥ 2 and their connectedness loci, known as multicorns. The multicorn M*subscript d is the connectedness locus in the space of antiholomorphic unicritical polynomials psubscript c(z) = ̄zsubscript d + c of degree d, i.e., the set of parameters for which the Julia set is connected. The special case d = 2 is the tricorn, which is the formal antiholomorphic analog to the Mandelbrot set.

BIFURCATION CURVES OF A FAMILY OF ONE-DIMENSIONAL BIQUADRATIC POLYNOMIAL MAPS, DEPENDING ON THE COMPLEX VARIABLE, AND TWO REAL PARAMETERS

Sun and Yin [2007] had presented a precise description of the connectedness locus of the family of real biquadratic polynomials {pa,b(z) = (z2 + a)2 + b}. We shall first give an elementary proof of their result. Second, we shall give a precise description of the sets of parameters (a, b) such that the family {pa,b} has attracting fixed points.

THE CONNECTEDNESS LOCUS OF A FAMILY OF REAL BIQUADRATIC POLYNOMIALS

In this paper we present a precise description of the connectedness locus of the family of polynomials (z2 + x)2 + y, where x, y are real numbers.