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A characterization of the kneading pair for bimodal degree one circle maps

Lluis Alsedà, Antonio Falcó (1997)

Annales de l'institut Fourier

For continuous maps on the interval with finitely many monotonicity intervals, the kneading theory developed by Milnor and Thurston gives a symbolic description of the dynamics of a given map. This description is given in terms of the kneading invariants which essentially consists in the symbolic orbits of the turning points of the map under consideration. Moreover, this theory also describes a classification of all such maps through theses invariants. For continuous bimodal degree one circle maps,...

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.

Characterization of ω -limit sets of continuous maps of the circle

David Pokluda (2002)

Commentationes Mathematicae Universitatis Carolinae

In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of ω -limit sets from the class 𝒞 ( I , I ) of continuous maps of the interval to the class 𝒞 ( 𝕊 , 𝕊 ) of continuous maps of the circle. Among others we give geometric characterization of ω -limit sets and then we prove that the family of ω -limit sets is closed with respect to the Hausdorff metric.

Codages de rotations et phénomènes d'autosimilarité

Boris Adamczewski (2002)

Journal de théorie des nombres de Bordeaux

Nous étudions une classe de suites symboliques, les codages de rotations, intervenant dans des problèmes de répartition des suites ( n α ) n ∈ ℕ et représentant une généralisation géométrique des suites sturmiennes. Nous montrons que ces suites peuvent être obtenues par itération de quatre substitutions définies sur un alphabet à trois lettres, puis en appliquant un morphisme de projection. L’ordre d’itération de ces applications est gouverné par un développement bi-dimensionnel de type “fraction continue”...

Combinatorics of distance doubling maps

Karsten Keller, Steffen Winter (2005)

Fundamenta Mathematicae

We study the combinatorics of distance doubling maps on the circle ℝ/ℤ with prototypes h(β) = 2β mod 1 and h̅(β) = -2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates f ∘ n of a distance doubling map f exhibit “distance doubling behavior”. The results include well known statements for h related to the structure of the Mandelbrot set M. For h̅ they suggest some analogies to the structure of...

Dynamics of circle maps with flat spots

Jacek Graczyk (2010)

Fundamenta Mathematicae

We study a certain class of weakly order preserving, non-invertible circle maps with irrational rotation numbers and exactly one flat interval. For this class of circle maps we explain the geometric and dynamic structure of orbits. In particular, we formulate the so called upper and lower scaling rules which show an asymmetric and double exponential decay of geometry.

Exact ∞ covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

We construct ∞ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ( n - 1 ∑ k = 0 n - 1 ν ∘ T - k ) n ≥ 1 of arithmetical averages of image measures does not converge weakly.

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