We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ 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.

We show that any diffeomorphism of a compact manifold can be $C^1$ approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.

We generalize the notion of topological pressure to the case of a finitely generated group of continuous maps and introduce group measure entropy. Also, we provide an elementary proof that any finitely generated group of polynomial growth admits a group invariant measure and show that for a group of polynomial growth its measure entropy is less than or equal to its topological entropy. The dynamical properties of groups of polynomial growth are reflected in the dynamics of some foliated spaces.

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology...

We prove that the Poincaré map ${\phi }_{(0,T)}$ has at least $N(\tilde{h},cl\left(W_0{W}_0^{-}\right))$ fixed points (whose trajectories are contained inside the segment W) where the homeomorphism $\tilde{h}$ is given by the segment W.

Given β > 1, let Tβ$\begin{array}{ccc}{\mathit{T}}_{\mathit{\beta }}\mathrm{:}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}& \mathrm{\to }& \mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}\\ \mathit{x}& \mathrm{\to }& \mathit{\beta x}\mathrm{-}\mathrm{\lfloor }\mathit{\beta x}\mathrm{\rfloor }\mathit{.}\end{array}$The iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of Tβ, where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion$\begin{array}{ccc}\mathit{x}\mathrm{=}\sum _{\mathit{i}\mathrm{\ge }\mathrm{0}}{\mathit{e}}_{\mathit{i}}{\mathit{\beta }}^{\mathrm{-}\mathit{i}}\mathit{,}& & \end{array}$where each ei...

The aim of this paper is to describe the set of periods of a Morse-Smale diffeomorphism of the two-dimensional sphere according to its homotopy class. The main tool for proving this is the Lefschetz fixed point theory.

Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let $f{̃}_{ϵ}$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $Fix\left(f{̃}_{ϵ}\right)=\varnothing$ for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó’s construction of Brouwer lines for orientation preserving homeomorphisms...

We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover, we show that both the density of such a measure and its entropy vary continuously with the parameter. In addition, we obtain exponential rate of mixing for these measures and also show that they satisfy the...