Parabolic orbifolds and the dimension of the maximal measure for rational maps.
Parapuzzle of the multibrot set and typical dynamics of unimodal maps
We study the parameter space of unicritical polynomials . For complex parameters, we prove that for Lebesgue almost every , the map is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every , the map is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.
Pasting together Julia sets: a worked out example of mating.
Periodic points and dynamic rays of exponential maps.
Periodic points of some algebraic maps.
Polynomial diffeomorphisms of : VII. Hyperbolicity and external rays
Polynomial diffeomorphisms of C2: currents, equilibrium measure and hyperbolicity.
Polynomial diffeomorphisms of C2. III. Ergodicity, exponents and entropy of the equilibrium measure.
Polynomial diffeomorphisms of C2. IV: The measure of maximal entropy and laminar currents.
Polynomials, Symmetry, and Dynamics: an Undertaking in Aesthetics
Porosity of Julia sets of non-recurrent and parabolic Collet-Eckmann rational functions.
Preperiodic dynatomic curves for
The preperiodic dynatomic curve is the closure in ℂ² of the set of (c,z) such that z is a preperiodic point of the polynomial with preperiod n and period p (n,p ≥ 1). We prove that each has exactly d-1 irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of . We also compute the genus of each component and the Galois group of the defining polynomial of .
Problems and solutions by the application of Julia set theory to one-dot and multi-dots numerical methods.
Puiseux series polynomial dynamics and iteration of complex cubic polynomials
We let be the completion of the field of formal Puiseux series and study polynomials with coefficients in as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in . We show that cubic polynomial dynamics over and are intimately related. More precisely, we establish that some elements of naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal...
Pulling back cohomology classes and dynamical degrees of monomial maps
We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.
Punti fissi di mappe olomorfe