scholarly journals Julia sets of complex Hénon maps

2018 ◽  
Vol 29 (07) ◽  
pp. 1850047
Author(s):  
Lorenzo Guerini ◽  
Han Peters
Keyword(s):  
Julia Set ◽  
A Priori ◽  
Natural Generalization ◽  
Dominated Splitting ◽  
Henon Maps ◽  
Additional Hypothesis ◽  
Hénon Maps ◽  
Open Questions ◽  
The One ◽  
Weaker Conditions

There are two natural definitions of the Julia set for complex Hénon maps: the sets [Formula: see text] and [Formula: see text]. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set [Formula: see text], under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used. We show that [Formula: see text] also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of [Formula: see text]. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on [Formula: see text]. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set [Formula: see text] were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].

Variation of topological pressure and dimension: from polynomials to complex Hénon maps

2013 ◽  
Vol 34 (3) ◽  
pp. 1018-1036
Author(s):  
CHRISTIAN WOLF
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Topological Pressure ◽  
Dimension Theory ◽  
Maximal Dimension ◽  
Pressure Function ◽  
Small Perturbations ◽  
Henon Maps ◽  
Hénon Maps ◽  
Regularity Results

AbstractWe study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

The Julia Set of Hénon Maps

2006 ◽  
Vol 334 (2) ◽  
pp. 457-464 ◽  
Author(s):  
John Erik Fornæss
Keyword(s):  
Julia Set ◽  
Henon Maps ◽  
Hénon Maps

scholarly journals Minkowski box from Yangian bootstrap

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Luke Corcoran ◽  
Florian Loebbert ◽  
Julian Miczajka ◽  
Matthias Staudacher
Keyword(s):  
Minkowski Space ◽  
Wigner Function ◽  
Functional Form ◽  
A Priori ◽  
Alternative Methods ◽  
Feynman Integrals ◽  
The One

Abstract We extend the recently developed Yangian bootstrap for Feynman integrals to Minkowski space, focusing on the case of the one-loop box integral. The space of Yangian invariants is spanned by the Bloch-Wigner function and its discontinuities. Using only input from symmetries, we constrain the functional form of the box integral in all 64 kinematic regions up to twelve (out of a priori 256) undetermined constants. These need to be fixed by other means. We do this explicitly, employing two alternative methods. This results in a novel compact formula for the box integral valid in all kinematic regions of Minkowski space.

scholarly journals Supergraph calculation of one-loop divergences in higher-derivative 6D SYM theory

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
I. L. Buchbinder ◽  
E. A. Ivanov ◽  
B. S. Merzlikin ◽  
K. V. Stepanyantz
Keyword(s):  
Effective Action ◽  
A Priori ◽  
Coupling Constant ◽  
Harmonic Superspace ◽  
Classical Action ◽  
Higher Derivative ◽  
Component Approach ◽  
Dimensionless Coupling ◽  
The One ◽  
Dimensionless Coupling Constant

Abstract We apply the harmonic superspace approach for calculating the divergent part of the one-loop effective action of renormalizable 6D, $$ \mathcal{N} $$ N = (1, 0) supersymmetric higher-derivative gauge theory with a dimensionless coupling constant. Our consideration uses the background superfield method allowing to carry out the analysis of the effective action in a manifestly gauge covariant and $$ \mathcal{N} $$ N = (1, 0) supersymmetric way. We exploit the regularization by dimensional reduction, in which the divergences are absorbed into a renormalization of the coupling constant. Having the expression for the one-loop divergences, we calculate the relevant β-function. Its sign is specified by the overall sign of the classical action which in higher-derivative theories is not fixed a priori. The result agrees with the earlier calculations in the component approach. The superfield calculation is simpler and provides possibilities for various generalizations.

scholarly journals Design of PDC Controllers by Matrix Reversibility for Synchronization ofYinandYangChaotic Takagi-Sugeno Fuzzy Henon Maps

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Yen Ho ◽  
Hsien-Keng Chen ◽  
Zheng-Ming Ge
Keyword(s):  
Chaos Synchronization ◽  
Matrix Theory ◽  
Chinese Philosophy ◽  
Fuzzy Model ◽  
Invertible Matrix ◽  
Henon Maps ◽  
Hénon Maps ◽  
Feminine Principle ◽  
Simulation Results ◽  
Takagi Sugeno

This paper investigates the synchronization ofYinandYangchaotic T-S fuzzy Henon maps via PDC controllers. Based on the Chinese philosophy,Yinis the decreasing, negative, historical, or feminine principle in nature, whileYangis the increasing, positive, contemporary, or masculine principle in nature.YinandYangare two fundamental opposites in Chinese philosophy. The Henon map is an invertible map; so the Henon maps with increasing and decreasing argument can be called theYangandYinHenon maps, respectively. Chaos synchronization ofYinandYangT-S fuzzy Henon maps is achieved by PDC controllers. The design of PDC controllers is based on the linear invertible matrix theory. The T-S fuzzy model ofYinandYangHenon maps and the design of PDC controllers are novel, and the simulation results show that the approach is effective.

scholarly journals Reversible Complex Hénon Maps

2002 ◽  
Vol 11 (3) ◽  
pp. 339-347 ◽  
Author(s):  
C. R. Jordan ◽  
D. A. Jordan ◽  
J. H. Jordan
Keyword(s):  
Henon Maps ◽  
Hénon Maps

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Richard A. Shore
Keyword(s):  
Reverse Mathematics ◽  
The Other ◽  
Base System ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Points Of View ◽  
The One ◽  
Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

Quasi-cyclic codes over the ring 𝔽p[u]/〈u2 − u〉

2015 ◽  
Vol 08 (04) ◽  
pp. 1550085
Author(s):  
Sukhamoy Pattanayak ◽  
Abhay Kumar Singh
Keyword(s):  
Structural Properties ◽  
Cyclic Codes ◽  
Natural Generalization ◽  
Spanning Set ◽  
The One ◽  
Quasi Cyclic Codes

Quasi-cyclic (QC) codes are a natural generalization of cyclic codes. In this paper, we study some structural properties of QC codes over [Formula: see text], where [Formula: see text] is a prime and [Formula: see text]. By exploring their structure, we determine the one generator QC codes over [Formula: see text] and the size by giving a minimal spanning set. We discuss some examples of QC codes of various length over [Formula: see text].

scholarly journals Generalized Hénon maps: the cubic diffeomorphisms of the plane

2000 ◽  
Vol 143 (1-4) ◽  
pp. 262-289 ◽  
Author(s):  
H.R. Dullin ◽  
J.D. Meiss
Keyword(s):  
Henon Maps ◽  
Hénon Maps

Three Kantian Routes to the Synthetic A Priori

2021 ◽  
pp. 1-28
Author(s):  
Robert Audi
Keyword(s):  
A Priori ◽  
Pure Reason ◽  
Critique Of Pure Reason ◽  
Insufficient Attention ◽  
Synthetic A Priori ◽  
The One ◽  
Philosophical Importance

Abstract Kant influentially distinguished analytic from synthetic a priori propositions, and he took certain propositions in the latter category to be of immense philosophical importance. His distinction between the analytic and the synthetic has been accepted by many and attacked by others; but despite its importance, a number of discussions of it since at least W. V. Quine’s have paid insufficient attention to some of the passages in which Kant draws the distinction. This paper seeks to clarify what appear to be three distinct conceptions of the analytic (and implicitly of the synthetic) that are presented in Kant’s Critique of Pure Reason and in some other Kantian texts. The conceptions are important in themselves, and their differences are significant even if they are extensionally equivalent. The paper is also aimed at showing how the proposed understanding of these conceptions—and especially the one that has received insufficient attention from philosophers—may bear on how we should conceive the synthetic a priori, in and beyond Kant’s own writings.

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