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Diffeomorphisms with weak shadowing

Kazuhiro Sakai (2001)

Fundamenta Mathematicae

The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C¹ interior of the set of all diffeomorphisms on a C ∞ closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C¹ interior is finite if and only if f is Morse-Smale.

Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation

Giuseppe Da Prato, Arnaud Debussche (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup P t ⁢ φ does exist for any bounded φ ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

For the real quadratic map P a ( x ) = x 2 + a and a given ϵ > 0 a point x has good expansion properties if any interval containing x also contains a neighborhood  J of x with P a n | J univalent, with bounded distortion and B ( 0 , ϵ ) ⊆ P a n ( J ) for some n ∈ ℕ . The ϵ -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [ α , - α ] . We show there is a set ℛ of parameters with positive Lebesgue measure for which the Hausdorff dimension of...

Dimensions des spirales

Yves Dupain, Michel Mendès France, Claude Tricot (1983)

Bulletin de la Société Mathématique de France

Distributional chaos on tree maps: the star case

Jose S. Cánovas (2001)

Commentationes Mathematicae Universitatis Carolinae

Let 𝕏 = { z ∈ ℂ : z n ∈ [ 0 , 1 ] } , n ∈ ℕ , and let f : 𝕏 → 𝕏 be a continuous map having the branching point fixed. We prove that f is distributionally chaotic iff the topological entropy of f is positive.

Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

Jeffrey Diller, Romain Dujardin, Vincent Guedj (2010)

Annales scientifiques de l'École Normale Supérieure

We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic “equilibrium” measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points...

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