we investigate the long-term behavior of solutions of the difference equation[ x_{n+1}=x_{n}x_{n-3}-1 ,, n=0 ,, 1 ,, ldots ,, ]noindent where the initial conditions $x_{-3} ,, x_{-2} ,, x_{-1} ,, x_{0}$ are real numbers.  in particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

Abstract A necessary and sufficient condition for the iterated function system $$\{f(\cdot ,\omega )\,|\,\omega \in \Omega \}$$ { f ( · , ω ) | ∈ Ω</mml...

Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, set of full measure, the pointwise This has Lorenz-like singularity and non-flat critical points.