About the attractive points of the root functions on the Riemannian foliation.
Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. ), and if μ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every “good” q, i.e. one for which “small neigh bourhoods arrive at large scale” under iteration of f. This generalizes the...
Accumulation constants of iterated function systems with Bloch target domains.
Algebraic degrees for iterates of meromorphic self-maps of Pk.
We first introduce the class of quasi-algebraically stable meromorphic maps of Pk. This class is strictly larger than that of algebraically stable meromorphic self-maps of Pk. Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.
An algebraic formulation of Thurston’s characterization of rational functions
Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston’s characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.
An algorithm for finding the Veech group of an origami.
An analogue to the Smale-Birkhoff homoclinic theorem for iterated entire mappings. (Summary).
An analogue to the Smale-Birkhoff homoclinic theorem for iterated entire mappings.
An application of the topological rigidity of the sine family.
Analytic invariants for the resonance
Associated to analytic Hamiltonian vector fields in having an equilibrium point satisfying a non semisimple resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.
Area and Hausdorff dimension of the set of accessible points of the Julia sets of λe^z and λ sin(z)
Asymptotic behaviors of complex analytic dynamical systems.
Attracting dynamics of exponential maps.