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Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and $h_{t o p} (T) > 0$ , it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.

Derivatives of circle rotations, and similarity of orbits.

Shutov, A.V. (2004)

Zapiski Nauchnykh Seminarov POMI

Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich

Raphaël Krikorian (2003/2004)

Séminaire Bourbaki

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension $2$ , il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à $2$ , et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

Different types of scaling in the dynamics of period-doubling maps under external periodic driving.

Ivan'kov, N.Yu., Kuznetsov, S.P. (2000)

Discrete Dynamics in Nature and Society

Differentiable rigidity and smooth conjugacy.

Jiang, Yunping (2005)

Annales Academiae Scientiarum Fennicae. Mathematica

Dimension groups for interval maps.

Shultz, Fred (2005)

The New York Journal of Mathematics [electronic only]

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

For the real quadratic map $P_{a} (x) = x^{2} + a$ and a given $ϵ > 0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_{a}^{n} |_{J}$ univalent, with bounded distortion and $B (0, ϵ) \subseteq P_{a}^{n} (J)$ for some $n \in ℕ$ . The $ϵ$ -weakly expanding set is the set of points which do not have good expansion properties. Let $α$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[α, - α]$ . We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff dimension of...

Diophantine classes, dimension and Denjoy maps

Bryna Kra, Jorg Schmeling (2002)

Acta Arithmetica

Discrete anisotropic curvature flow of graphs

Klaus Deckelnick, Gerhard Dziuk (1999)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Discrete anisotropic curvature flow of graphs

Klaus Deckelnick, Gerhard Dziuk (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfaces in natural norms. Furthermore we prove a global existence result for the continuous problem and present some examples of computed surfaces.

Dispersion properties of ergodic translations.

Isola, Stefano (2006)

International Journal of Mathematics and Mathematical Sciences

Distortion bounds for $C^{2 + η}$ unimodal maps

Mike Todd (2007)

Fundamenta Mathematicae

We obtain estimates for derivative and cross-ratio distortion for $C^{2 + η}$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_{T} : T \to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove that for...

Distributional chaos for flows

Yunhua Zhou (2013)

Czechoslovak Mathematical Journal

Schweizer and Smítal introduced the distributional chaos for continuous maps of the interval in B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344 (1994), 737–854. In this paper, we discuss the distributional chaos DC1–DC3 for flows on compact metric spaces. We prove that both the distributional chaos DC1 and DC2 of a flow are equivalent to the time-1 maps and so some properties of DC1 and DC2 for discrete systems...

Distributional chaos on tree maps: the star case

Jose S. Cánovas (2001)

Commentationes Mathematicae Universitatis Carolinae

Let $𝕏 = {z \in ℂ : z^{n} \in [0, 1]}$ , $n \in ℕ$ , and let $f : 𝕏 \to 𝕏$ be a continuous map having the branching point fixed. We prove that $f$ is distributionally chaotic iff the topological entropy of $f$ is positive.

Does a billiard orbit determine its (polygonal) table?

Jozef Bobok, Serge Troubetzkoy (2011)

Fundamenta Mathematicae

We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation under additional regularity conditions on the orbit.

Dynamically defined Cantor sets under the conditions of McDuff's conjecture

Jorge Iglesias, Aldo Portela (2010)

Colloquium Mathematicae

We prove that if the Cantor set K, dynamically defined by a function $S \in C^{1 + α}$ , satisfies the conditions of McDuff’s conjecture then it cannot be C¹-minimal.

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