Entire Mappings Research Articles

Voluntary task switching is affected by modality compatibility and preparation

Cognitive task control can be examined in task-switching studies. Performance costs in task switches are usually smaller with compatible stimulus-response modality mappings (visual-manual and auditory-vocal) than with incompatible mappings (visual-vocal and auditory-manual). Modality compatibility describes the modality match of sensory input and of the anticipated response effect (e.g., vocal responses produce auditory effects, so that auditory stimuli are modality-compatible with vocal responses). Fintor et al. (Psychological Research, 84(2), 380–388, 2020) found that modality compatibility also biased task choice rates in voluntary task switching (VTS). In that study, in each trial participants were presented with a visual or auditory spatial stimulus and were free to choose the response modality (manual vs. vocal). In this free-choice task, participants showed a bias to create more modality-compatible than -incompatible mappings. In the present study, we assessed the generality of Fintor et al.’s (2020) findings, using verbal rather than spatial stimuli, and more complex tasks, featuring an increased number of stimulus-response alternatives. Experiment 1 replicated the task-choice bias to preferentially create modality-compatible mappings. We also found a bias to repeat the response modality just performed, and a bias to repeat entire stimulus-response modality mappings. In Experiment 2, we manipulated the response-stimulus interval (RSI) to examine whether more time for proactive cognitive control would help resolve modality-specific crosstalk in this free-choice paradigm. Long RSIs led to a decreased response-modality repetition bias and mapping repetition bias, but the modality-compatibility bias was unaffected. Together, the findings suggest that modality-specific priming of response modality influences task choice.

On harmonic entire mappings II

On harmonic entire mappings II

On harmonic entire mappings

In this paper, we investigate properties of harmonic entire mappings. Firstly, we give the characterizations of the order and the type for a harmonic entire mapping \(f=h+\overline{g}\), respectively, and also consider the relationship between the order and the type of f, h, and g. Secondly, we investigate the harmonic mappings \(f=h+\overline{g}\) such that \(f^{(n_p)}=h^{(n_p)}+\overline{g^{(n_p)}}\) are univalent in the unit disk, where \(\{n_p\}_{p=1}^{\infty }\) be a strictly increasing sequence of nonnegative integers. In terms of the sequence \(\{n_p\}_{p=1}^{\infty }\), we derive several necessary conditions for these mappings to be entire and also establish an upper bound for the order of these mappings.

The Composition Operation on Spaces of Holomorphic Mappings

AbstractWe discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

Some aspects of shift-like automorphisms of $$\pmb {\mathbb {C}}^{\varvec{k}}$$ C k

The goal of this article is two fold. First, using transcendental shift-like automorphisms of \(\mathbb C^k, ~k \ge 3\) we construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of \(\mathbb C^k, ~k \ge 3\) whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon–Esterle in \(\mathbb C^2.\) The second example shows the existence of a Fatou–Bieberbach domain in \(\mathbb C^k,~k \ge 3\) that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay–Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in ℝn, n ⩾ 3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in ℝ3, while finite, can be much larger than possible in the plane.

Mappings of Bounded Mean Distortion and Cohomology

We obtain a quantitative cohomological boundedness theorem for closed manifolds receiving entire mappings of bounded mean distortion and finite lower order. We also prove an equidistribution theorem for mappings of finite distortion.

Factorization of entire mappings of nuclear bounded type

In this paper we study the relationships between the spaces of entire mappings of bounded type, entire mappings of nuclear bounded type, entire mappings of Pietsch integral bounded type, and entire mappings of Grothendieck integral bounded type. Several results due to Alencar ( Proc. Roy. Irish Acad. 85A (1985) 131–138) and Cilia and Gutiérrez ( J. Aust. Math. Soc. 76 (2004) 269–280) for homogeneous polynomials are extended to entire mappings. In the main result we prove that an entire mapping is of nuclear bounded type if and only if it factors through an entire mapping of Pietsch integral bounded type.

Homology of real algebraic varieties and morphisms to spheres

Let X and Y be affine nonsingular real algebraic varieties. One of the classical problems in real algebraic geometry is whether a given C ∞ mapping f: X → Y can be approximated by entire rational mappings in the space of C ∞ mappings. In this work, we obtain some sufficient conditions in the case when Y is the standard sphere S n .

On Köthe sequence spaces and linear logic

We present a category of locally convex topological vector spaces that is a model of propositional classical linear logic and is based on the standard concept of Köthe sequence spaces. In this setting, the ‘of course’ connective of linear logic has a quite simple structure of a commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting in which typed λ-calculus and differential calculus can be combined; we give a few examples of computations.

Asymptotic rigidity of scaling ratios for critical circle mappings

Let $f$ be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of $f$ about the critical point are asymptotically independent of $f$. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of $f$ to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for $f$ is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.

Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and let C n denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C n →C n and for holomorphic automorphisms of C n on discrete subsets of C n.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C n.For each closed complex submanifold (or subvariety) M ⊂ C n of complex dimension m < n we construct a domain Ω ⊂C n containing M and a biholomorphic map F: Ω → C n onto C n with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C n−m →C n at infinitely many points. If m = n − 1, we construct F as above such that C n ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C m →C m−1 such that the complement C m+1 ∖F(C m )is hyperbolic.

Fatou and Julia sets for entire mappings in ${\Bbb C}^k$

the Fatou set is just made of basins of attraction correspondingto attracting cycles and that hyperbolic periodic cycles are dense in the Julia set.Similar results are obtained in ([FS3]), ([FS4]) for other classes of mappings, seeTheorems 2.9 and 2.10 below.In ([Bu]), Buzzard proved a version of the Kupka-Smale Theorem for

On the growth of two-dimensional quasiregular entire mappings

In the present paper we study two-dimensional quasiregular entire mappings from the point of view of growth. By using the Stoilow decomposition theorem, we establish inequalities on the order, type and class of such a mapping. A special attention is paid to mappings of order zero by means of the functional τ.

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouche to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.

Entire Functions of Bounded Type on Fréchet Spaces

We show that holomorphic mappings of bounded type defined on Frechet spaces extend to the bidual. The relationship between holomorphic mappings of bounded type and of uniformly bounded type is discussed and some algebraic and topological properties of the space of all entire mappings of (uniformly) bounded type are proved, for example a holomorphic version of Schauder's theorem.

Snapback repellers and semiconjugacy for iterated entire mappings

A theorem of Poincare guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we related the multiplicity of the zeros of the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two.

On Picard's Theorem for Entire Quasiregular Mappings

On Picard's Theorem for Entire Quasiregular Mappings

On Picard’s theorem for entire quasiregular mappings

Several refinements of Picard’s theorem for entire functions in the complex plane have been proved by many authors in connection with the theory of Picard sets. We prove a result of this type for entire quasiregular mappings in euclidean n n -space in the case when the "Picard set" consists of a sequence a k {a_k} on a ray emanating from 0 with | a k | = 2 k |{a_k}| = {2^k} .