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A Case of Monotone Ratio Growth for Quadratic-Like Mappings

Waldemar Pałuba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

This is a study of the monotone (in parameter) behavior of the ratios of the consecutive intervals in the nested family of intervals delimited by the itinerary of a critical point. We consider a one-parameter power-law family of mappings of the form f a = - | x | α + a . Here we treat the dynamically simplest situation, before the critical point itself becomes strongly attracting; this corresponds to the kneading sequence RRR..., or-in the quadratic family-to the parameters c ∈ [-1,0] in the Mandelbrot set. We allow...

A characterization of ω-limit sets for piecewise monotone maps of the interval

Andrew D. Barwell (2010)

Fundamenta Mathematicae

For a piecewise monotone map f on a compact interval I, we characterize the ω-limit sets that are bounded away from the post-critical points of f. If the pre-critical points of f are dense, for example when f is locally eventually onto, and Λ ⊂ I is closed, invariant and contains no post-critical point, then Λ is the ω-limit set of a point in I if and only if Λ is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of...

A classification of inverse limit spaces of tent maps with periodic critical points

Lois Kailhofer (2003)

Fundamenta Mathematicae

We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps f a , f b with periodic critical points, we show that the inverse limit spaces ( a , f a ) and ( b , g b ) are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.

A general class of iterative equations on the unit circle

Marek Cezary Zdun, Wei Nian Zhang (2007)

Czechoslovak Mathematical Journal

A class of functional equations with nonlinear iterates is discussed on the unit circle 𝕋 1 . By lifting maps on 𝕋 1 and maps on the torus 𝕋 n to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.

A topological invariant for pairs of maps

Marcelo Polezzi, Claudemir Aniz (2006)

Open Mathematics

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝn, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘...

Adding machines, endpoints, and inverse limit spaces

Lori Alvin, Karen Brucks (2010)

Fundamenta Mathematicae

Let f be a unimodal map in the logistic or symmetric tent family whose restriction to the omega limit set of the turning point is topologically conjugate to an adding machine. A combinatoric characterization is provided for endpoints of the inverse limit space (I,f), where I denotes the core of the map.

Attractors and Inverse Limits.

James Keesling (2008)

RACSAM

This paper surveys some recent results concerning inverse limits of tent maps. The survey concentrates on Ingram’s Conjecture. Some motivation is given for the study of such inverse limits.

Breaking the continuity of a piecewise linear map

Viktor Avrutin, Michael Schanz, Björn Schenke (2012)

ESAIM: Proceedings

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...

Central limit theorem for sampled sums of dependent random variables

Nadine Guillotin-Plantard, Clémentine Prieur (2010)

ESAIM: Probability and Statistics

We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to dependent random variables sampled by a ℤ -valued transient random walk. This extends the results obtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics3 (2003) 477–497]. An application to parametric estimation by random sampling is also provided.

Centralisateurs des difféomorphismes de la demi-droite

Hélène Eynard-Bontemps (2008/2009)

Séminaire de théorie spectrale et géométrie

Soit f un difféomorphisme lisse de ℝ + fixant seulement l’origine, et 𝒵 r son centralisateur dans le groupe des difféomorphismes 𝒞 r . Des résultat classiques de Kopell et Szekeres montrent que 𝒵 1 est toujours un groupe à un paramètre. En revanche, Sergeraert a construit un f dont le centralisateur 𝒵 ∞ est réduit au groupe des itérés de f . On présente ici le résultat principal de [3] : 𝒵 ∞ peut en fait être un sous-groupe propre et non-dénombrable (donc dense) de 𝒵 1 .

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