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The continuous solutions of a generalized Dhombres functional equation

L. ReichJaroslav SmítalM. Štefánková — 2004

Mathematica Bohemica

We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J J is a given increasing homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) J is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under ϕ and which contains in its interior no fixed point except for 1 . They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

The converse problem for a generalized Dhombres functional equation

L. ReichJaroslav SmítalM. Štefánková — 2005

Mathematica Bohemica

We consider the functional equation f ( x f ( x ) ) = ϕ ( f ( x ) ) where ϕ J J is a given homeomorphism of an open interval J ( 0 , ) and f ( 0 , ) J is an unknown continuous function. A characterization of the class 𝒮 ( J , ϕ ) of continuous solutions f is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f ( 0 , ) J , where J is an interval, there is an increasing homeomorphism ϕ of J such that f 𝒮 ( J , ϕ ) . We...

Gibbs-Markov-Young structures, ,

Carla L. Dias — 2012

ESAIM: Proceedings

We discuss the geometric structures defined by Young in [9, 10], which are used to prove the existence of an ergodic absolutely continuous invariant probability measure and to study the decay of correlations in expanding or hyperbolic systems on large parts.

Coexisting cycles in a class of 3-D discrete maps

Anna Agliari — 2012

ESAIM: Proceedings

In this paper we consider the class of three-dimensional discrete maps () = [(), (), ()], where : ℝ → ℝ is an endomorphism. We show that all the cycles of the 3-D map can be obtained by those of (), as well as their local bifurcations. In particular we obtain that any local bifurcation is of co-dimension 3, that is three eigenvalues cross simultaneously the unit circle. As the map exhibits coexistence of cycles when ...

Some aspects of the local theory of generalized Dhombres functional equations in the complex domain

Jörg Tomaschek — 2012

ESAIM: Proceedings

We study the generalized Dhombres functional equation (()) = (()) in the complex domain. The function is given and we are looking for solutions with (0) =  and is a primitive root of unity of order  ≥ 2. All formal solutions for this case are described in this work, for the situation where can be transformed into a function which is linearizable and local analytic in a neighbourhood of zero...

On gradient-like random dynamical systems

Aya HmissiFarida HmissiMohamed Hmissi — 2012

ESAIM: Proceedings

This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS. The obtained results are generalizations of well-known analogous theorems in the framework of deterministic...

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