Forewords : European Conference on Iteration Theory - ECIT 2010
Some remarks on the general theorem of the existence of iterative roots of homeomorphisms with a rational rotation number
We show that the theorem proved in [8] generalises the previous results concerning orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number (see [2], [6], [10] and [7]).
Invariant graphs of functions for the mean-type mappings
Let
Gibbs-Markov-Young structures, ,
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
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
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...
Example of the Smooth Skew Product in the Plane with the One-dimensional Ramified Continuum as the Global Attractor
The example is constructed of the -smooth skew product of interval maps possessing the one-dimensional ramified continuum (containing no arcs homeomorphic to the circle) with an infinite set of ramification points as the global attractor.
Doubling bifurcation of a closed invariant curve in 3D maps
The object of the present paper is to give a qualitative description of the bifurcation mechanisms associated with a closed invariant curve in three-dimensional maps, leading to its doubling, not related to a standard doubling of tori. We propose an explanation on how a closed invariant attracting curve, born via Neimark-Sacker bifurcation, can be transformed into a repelling one giving birth to a new attracting closed invariant curve which has doubled...
On gradient-like random dynamical systems
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...
On the helix equation
This paper is devoted to the helices processes, i.e. the solutions
Periodicity of -expansions for certain Pisot units
Given
Symbolic dynamics and Lyapunov exponents for Lozi maps
Building on the kneading theory for Lozi maps introduced by Yutaka Ishii, in 1997, we introduce a symbolic method to compute its largest Lyapunov exponent. We use this method to study the behavior of the largest Lyapunov exponent for the set of points whose forward and backward orbits remain bounded, and find the maximum value that the largest Lyapunov exponent can assume.
On evolution of small spheres in the phase space of a dynamical system
We study the connection between the entropy of a dynamical system and the boundary distortion rate of regions in the phase space of the system.
On the formal first cocycle equation for iteration groups of type II
Let
Breaking the continuity of a piecewise linear map
Knowledge about the behavior of discontinuous piecewise-linear maps is important for a wide range of applications. An efficient way to investigate the bifurcation structure in 2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points, called organizing centers, and to describe the bifurcation structure in their neighborhood. In this work, we present the organizing centers in the 1D discontinuous piecewise-linear map...
Exponential wealth distribution : a new approach from functional iteration theory
Different approaches are possible in order to derive the exponential regime in statistical systems. Here, a new functional equation is proposed in an economic context to explain the wealth exponential distribution. Concretely, the new iteration [1] given by It is found that the exponential distribution is a stable fixed point of this functional iteration equation. From this point of view, it is easily understood why the exponential wealth distribution...
Substitution systems associated with the dynamical system (𝒜, )
We consider the dynamical system (𝒜, ), where 𝒜 is a class of differential real functions defined on some interval and : 𝒜 → 𝒜 is an operator := , where is a differentiable -modal map. If we consider functions in 𝒜 whose critical values are periodic points for then, we show how to define and characterize a substitution system associated with (𝒜, ). For...
Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map
In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns...
Organizing centers in parameter space of discontinuous 1D maps. The case of increasing/decreasing branches
This work contributes to classify the dynamic behaviors of piecewise smooth systems in which characterize the qualitative changes in the dynamics. A central point of our investigation is the intersection of two border collision bifurcation curves in a parameter plane. This problem is also associated with the continuity breaking in a fixed point of a piecewise smooth map. We will relax the hypothesis needed in [4] where it was proved that in the case...
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