On the transitive and $\omega $-limit points of the continuous mappings of the circle

David Pokluda

Displaying similar documents to “On the transitive and $ω$ -limit points of the continuous mappings of the circle”

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

David Pokluda (2002)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. (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.

Sets of nondifferentiability for conjugacies between expanding interval maps

Thomas Jordan, Marc Kesseböhmer, Mark Pollicott, Bernd O. Stratmann (2009)

Fundamenta Mathematicae

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We study differentiability of topological conjugacies between expanding piecewise $C^{1 + ϵ}$ interval maps. If these conjugacies are not C¹, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the...

The converse problem for a generalized Dhombres functional equation

L. Reich, Jaroslav Smítal, M. Štefánková (2005)

Mathematica Bohemica

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We consider the functional equation $f (x f (x)) = ϕ (f (x))$ where $ϕ J \to J$ is a given homeomorphism of an open interval $J \subset (0, \infty)$ and $f (0, \infty) \to J$ 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 $f (0, \infty) \to J$ , where $J$ is an interval, there is an increasing homeomorphism $ϕ$ of $J$ such...

Kneading sequences for double standard maps

Michael Benedicks, Ana Rodrigues (2009)

Fundamenta Mathematicae

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We investigate the symbolic dynamics for the double standard maps of the circle onto itself, given by $f_{a, b} (x) = 2 x + a + (b / π) s i n (2 π x) (m o d 1)$ , where b = 1 and a is a real parameter, 0 ≤ a < 1.

The structure of $ω$ -limit sets for continuous maps of the interval

Andrew M. Bruckner, Jaroslav Smítal (1992)

Mathematica Bohemica

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We prove that every infinite nowhere dense compact subset of the interval $I$ is an $ω$ -limit set of homoclinic type for a continuous function from $I$ to $I$ .

Dense chaos

Ľubomír Snoha (1992)

Commentationes Mathematicae Universitatis Carolinae

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According to A. Lasota, a continuous function $f$ from a real compact interval $I$ into itself is called generically chaotic if the set of all points $(x, y)$ , for which ${lim inf}_{n \to \infty} | f^{n} (x) - f^{n} (y) | = 0$ and ${lim sup}_{n \to \infty} | f^{n} (x) - f^{n} (y) | > 0$ , is residual in $I \times I$ . Being inspired by this definition we say that $f$ is densely chaotic if this set is dense in $I \times I$ . A characterization of the generically chaotic functions is known. In the paper the densely chaotic functions are characterized and it is proved that in the class of piecewise monotone maps with finite number...

Diagonal points having dense orbit

T. K. Subrahmonian Moothathu (2010)

Colloquium Mathematicae

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Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying...

Displaying similar documents to “On the transitive and ω -limit points of the continuous mappings of the circle”

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

Sets of nondifferentiability for conjugacies between expanding interval maps

The converse problem for a generalized Dhombres functional equation

Kneading sequences for double standard maps

The structure of ω -limit sets for continuous maps of the interval

Dense chaos

Diagonal points having dense orbit

Displaying similar documents to “On the transitive and $ω$ -limit points of the continuous mappings of the circle”

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

The structure of $ω$ -limit sets for continuous maps of the interval