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

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.

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Julia sets of complex Hénon maps

International Journal of Mathematics ◽

10.1142/s0129167x18500477 ◽

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)].

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Bowen’s formula for meromorphic functions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711000290 ◽

2011 ◽

Vol 32 (4) ◽

pp. 1165-1189 ◽

Cited By ~ 2

Author(s):

KRZYSZTOF BARAŃSKI ◽

BOGUSŁAWA KARPIŃSKA ◽

ANNA ZDUNIK

Keyword(s):

Hausdorff Dimension ◽

Meromorphic Function ◽

Meromorphic Functions ◽

Julia Set ◽

Topological Pressure ◽

Singular Set ◽

Dimension Zero ◽

Common Value ◽

Bounded Set ◽

The Common

AbstractLet f be an arbitrary transcendental entire or meromorphic function in the class 𝒮 (i.e. with finitely many singularities). We show that the topological pressure P(f,t) for t>0 can be defined as the common value of the pressures P(f,t,z) for all z∈ℂ up to a set of Hausdorff dimension zero. Moreover, we prove that P(f,t) equals the supremum of the pressures of f∣X over all invariant hyperbolic subsets X of the Julia set, and we prove Bowen’s formula for f, i.e. we show that the Hausdorff dimension of the radial Julia set of f is equal to the infimum of the set of t, for which P(f,t) is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions f in the class ℬ (i.e. with a bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

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The Julia Set of Hénon Maps

Mathematische Annalen ◽

10.1007/s00208-005-0743-2 ◽

2006 ◽

Vol 334 (2) ◽

pp. 457-464 ◽

Cited By ~ 7

Author(s):

John Erik Fornæss

Keyword(s):

Julia Set ◽

Henon Maps ◽

Hénon Maps

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

Abstract and Applied Analysis ◽

10.1155/2012/358201 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11 ◽

Cited By ~ 1

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.

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