On unimodal maps with critical order 2 + ε
It is proved that a smooth unimodal interval map with critical order 2 + ε has no wild attractor if ε >0 is small.
Dimensions of the Julia sets of rational maps with the backward contraction property
Consider a rational map f on the Riemann sphere of degree at least 2 which has no parabolic periodic points. Assuming that f has Rivera-Letelier's backward contraction property with an arbitrarily large constant, we show that the upper box dimension of the Julia set J(f) is equal to its hyperbolic dimension, by investigating the properties of conformal measures on the Julia set.
Real Koebe principle
We prove a version of the real Koebe principle for interval (or circle) maps with non-flat critical points.
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.
Existence of unique SRB-measures is typical for real unicritical polynomial families
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