Periodic points and smooth rays

Let P : C → C P: \mathbb {C} \to \mathbb {C} be a polynomial map with disconnected filled Julia set K P K_P and let z 0 z_0 be a repelling or parabolic periodic point of P P . We show that if the connected component of K P K_P containing z 0 z_0 is non-degenerate, then z 0 z_0 is the landing point of at least one smooth external ray. The statement is optimal in the sense that all but one cycle of rays landing at z 0 z_0 may be broken.

IMAGE ENCRYPTION BASED ON TWO-DIMENSIONAL FRACTIONAL QUADRIC POLYNOMIAL MAP

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.

On Some Symmetries of Quadratic Systems

We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented.

ON THE CLASSIFICATION BY MORIMOTO AND NAGANO

We consider a family $M_{t}^{3}$ , with $t>1$ , of real hypersurfaces in a complex affine three-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in $\mathbb{C}^{n}$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the Cauchy–Riemann (CR)-embeddability of $M_{t}^{3}$ in $\mathbb{C}^{3}$ . In our earlier article, we showed that $M_{t}^{3}$ is CR-embeddable in $\mathbb{C}^{3}$ for all $1<t<\sqrt{(2+\sqrt{2})/3}$ . In the present paper, we prove that $M_{t}^{3}$ can be immersed in $\mathbb{C}^{3}$ for every $t>1$ by means of a polynomial map. In addition, one of the immersions that we construct helps simplify the proof of the above CR-embeddability theorem and extend it to the larger parameter range $1<t<\sqrt{5}/2$ .

Polynomial shift-like maps in ℂk

The purpose of this paper is to explore a few properties of polynomial shift-like automorphisms of [Formula: see text] We first prove that a [Formula: see text]-shift-like polynomial map (say [Formula: see text]) degenerates essentially to a polynomial map in [Formula: see text]-dimensions as [Formula: see text] Second, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e. [Formula: see text], where [Formula: see text] is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

Quantitative properties of the non-properness set of a polynomial map, a positive characteristic case

Let [Formula: see text] be a generically finite polynomial map of degree [Formula: see text] between affine spaces. In [Z. Jelonek and M. Lasoń, Quantitative properties of the non-properness set of a polynomial map, Manuscripta Math. 156(3–4) (2018) 383–397] we proved that if [Formula: see text] is the field of complex or real numbers, then the set [Formula: see text] of points at which [Formula: see text] is not proper is covered by polynomial curves of degree at most [Formula: see text]. In this paper, we generalize this result to positive characteristic. We provide a geometric proof of an upper bound by [Formula: see text].

Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values

Let $$X\subset {\mathbb {C}}^n$$ X ⊂ C n be an affine variety and $$f:X\rightarrow {\mathbb {C}}^m$$ f : X → C m be the restriction to X of a polynomial map $${\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$$ C n → C m . We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of $${\mathbb {C}}^m$$ C m which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.