Unstable manifolds and Hölder structures associated with noninvertible maps

Mihăilescu, Eugen

doi:10.3934/dcds.2006.14.419

Cited by 26 publications

(36 citation statements)

References 24 publications

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“…In this paper, we are interested to apply the results in [6] and [7] to study a large class of maps obtained as perturbations of (z 2 , w 2 ) and to identify a set of elements of this class which are injective on their respective basic sets, although these polynomial maps are not injective on C 2 .…”

Section: Introductionmentioning

confidence: 99%

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Dynamical behavior of endomorphisms on certain invariant sets

Putan¹,

Stan²

2013

Mathematica Slovaca

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ABSTRACT. We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space P 2 . We consider the perturbation (z 2 + εz + bεw 2 , w 2 ) of (z 2 , w 2 ) and we prove that, for b sufficiently small, it is injective on its basic set Λ ε close to Λ := {0} × S 1 . Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ ε , in the case of these maps.

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Section: Introductionmentioning

confidence: 99%

“…Then, in [4], Manning and McCluskey applied thermodynamical formalism to formulas for the Hausdorff dimension in the case of diffeomorphisms on surfaces. The situation for non-invertible higher-dimensional maps is different and was studied in [5], [6], [7] and [8].…”

Section: Introductionmentioning

confidence: 99%

Dynamical behavior of endomorphisms on certain invariant sets

Putan¹,

Stan²

2013

Mathematica Slovaca

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“…We use a standard notation, namely Df s (x) := Df | E s x , x ∈ Λ and Df u (x) := Df | E û x ,x ∈Λ. Examples and properties of endomorphisms with some hyperbolicity have been studied by many authors, for instance [4], [11], [16], [18], [6], [19], [7], [9], etc. Notice that if Λ is a repeller, then the local stable manifolds are contained in Λ. Hyperbolicity on Λ assures the existence of a unique equilibrium (Gibbs) measure µ φ for any given Hölder continuous potential φ on Λ; equilibrium measures are of great interest and have been studied intensively in the literature (for instance [17], [1], [14], [3], [5], [9], etc.…”

Section: Introductionmentioning

confidence: 99%

“…. , ) with [16], [9]. Also the local unstable manifolds depend on backward trajectories; there is some r > 0 and local stable and local unstable manifolds, W s r (x) and W u r (x) for anyx ∈Λ.…”

Section: Introductionmentioning

confidence: 99%

“…Sometimes we will call such repellers, folded repellers to emphasize that the map is not necessarily invertible on them. We shall work in the sequel with hyperbolic saddle-type repellers, i.e f is assumed to be hyperbolic as a non-invertible map on Λ (see [16], [9], etc.) and it has both stable and unstable directions in the tangent bundle; the map f is not assumed to be expanding on Λ.…”

Section: Introductionmentioning

confidence: 99%

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Entropy Production for a Class of Inverse SRB Measures

Mihăilescu

Urbański

2012

J Stat Phys

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Dynamics in Several Complex Variables

Abate

2017

Lecture Notes in Mathematics

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The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings.The first chapter deals with holomorphic endomorphisms of the projective space P k . We establish the first properties and give several constructions for the Green currents T p and the equilibrium measure µ = T k . The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E , totally invariant, i.e. f −1 (E ) = f (E ) = E , such that when a ∈ E , the probability measures, equidistributed on the fibers f −n (a), converge towards the equilibrium measure µ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced by varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of µ: Kmixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(P k ) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. µ, e α|ϕ| ≤ c, on bounded sets of ϕ in DSH(P k ), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of µ.The second chapter develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of C k with V convex and U ⋐ V . We introduce the dynamical degrees for such maps and construct the equilibrium measure µ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure µ. The assumption is stable under small pertubations on the map. We also study the dimension of µ, the Lyapounov exponents and their variation.Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

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Unstable manifolds and Hölder structures associated with noninvertible maps

Cited by 26 publications

References 24 publications

Dynamical behavior of endomorphisms on certain invariant sets

Dynamical behavior of endomorphisms on certain invariant sets

Entropy Production for a Class of Inverse SRB Measures

Dynamics in Several Complex Variables

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