Equicontinuity of iterates of circle maps.
Faà di Bruno's formula and nonhyperbolic fixed points of one-dimensional maps.
Fixed points and iterations of mean-type mappings
Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping...
Forcing relation on minimal interval patterns
Let ℳ be the set of pairs (T,g) such that T ⊂ ℝ is compact, g: T → T is continuous, g is minimal on T and has a piecewise monotone extension to convT. Two pairs (T,g),(S,f) from ℳ are equivalent if the map h: orb(minT,g) → orb(minS,f) defined for each m ∈ ℕ₀ by is increasing on orb(minT,g). An equivalence class of this relation-a minimal (oriented) pattern A-is exhibited by a continuous interval map f:I → I if there is a set T ⊂ I such that (T,f|T) = (T,f) ∈ A. We define the forcing relation on...
Generalizing the arithmetic geometric mean - a hapless computer experiment.
Generic chaos
Generic Properties of Continuous Iterated Function Systems with Place Dependent Probabilities
It is shown that for a typical continuous learning system defined on a compact convex subset of ℝⁿ the Hausdorff dimension of its invariant measure is equal to zero.
Hardy fields and existence of transexponential functions.
Infinite discrete sets of limit points of iterative sequences of contiunous functions.
Infinite discrete sets of limit points of iterative sequences of continuous functions. (Short Communication).
Inhomogeneous self-similar sets and box dimensions
We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than upper box dimension, Hausdorff dimension and packing dimension.
Invariance identity in the class of generalized quasiarithmetic means
An invariance formula in the class of generalized p-variable quasiarithmetic means is provided. An effective form of the limit of the sequence of iterates of mean-type mappings of this type is given. An application to determining functions which are invariant with respect to generalized quasiarithmetic mean-type mappings is presented.
Invariant graphs of functions for the mean-type mappings
Let I be a real interval, J a subinterval of I, p ≥ 2 an integer number, and M1, ... , Mp : Ip → I the continuous means. We consider the problem of invariance of the graphs of functions ϕ : Jp−1 → I with respect to the mean-type mapping M = (M1, ... , Mp).Applying a result on the existence and uniqueness of an M -invariant mean [7], we prove that if the graph of a continuous function ϕ : Jp−1 → I ...
Invariant scrambled sets and maximal distributional chaos
For the full shift (Σ₂,σ) on two symbols, we construct an invariant distributionally ϵ-scrambled set for all 0 < ϵ < diam Σ₂ in which each point is transitive, but not weakly almost periodic.
Iterated quasi-arithmetic mean-type mappings
We work with a fixed N-tuple of quasi-arithmetic means generated by an N-tuple of continuous monotone functions (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping tend pointwise to a mapping having values on the diagonal of . Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means taken on b. We effectively...
Iterates of piecewise monotonic continuous functions
Iteration groups generated by functions
Iteration semigroups with generalized convex, concave and affine elements.
La suite des approximations successives dans le cas general
Maps of the interval Ljapunov stable on the set of nonwandering points.