### A counterexample to a Fedorenko statement.

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In the paper [CEGHM] a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension n ≥ 3 are given. In the present paper we explain a way for obtaining a family of polynomial counterexamples containing the above ones. Finally we study the global dynamics of the examples given in [CEGHM].

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. ${f}^{-1}\left(A\right)=A$), 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...

We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if ${p}_{1}<...<{p}_{n}$ is a periodic orbit of a continuous map f then there is a union set ${q}_{1},...,{q}_{n-1}$ of some periodic orbits of f such that ${p}_{i}<{q}_{i}<{p}_{i+1}$ for any i.

We consider a chaotic system with a double-scroll attractor proposed by Elwakil, composing with a second-order system, which has low-dimensional multiple invariant subspaces and multi-level on-off intermittency. This type of composite system always includes a skew-product structure and some invariant subspaces, which are associated with different levels of laminar phase. In order for the level of laminar phase be adjustable, we adopt a nonlinear function with saturation characteristic to tune the...

Let F be a germ of analytic transformation of (Cp, 0). We say that F is semi-attractive at the origin, if F'(0) has one eigenvalue equal to 1 and if the other ones are of modulus strictly less than 1. The main result is: either there exists a curve of fixed points, or F - Id has multiplicity k and there exists a domain of attraction with k - 1 petals. We also study the case where F is a global isomorphism of C2 and F - Id has multiplicity k at the origin. This work has been inspired by two papers:...