### The $n$th root of a braid is unique up to conjugacy.

Gonzalez-Meneses, Juan (2003)

Algebraic & Geometric Topology

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Gonzalez-Meneses, Juan (2003)

Algebraic & Geometric Topology

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Boju Jiang (1999)

Banach Center Publications

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In this talk, we shall look at the application of Nielsen theory to certain questions concerning the "homotopy minimum" or "homotopy stability" of periodic orbits under deformations of the dynamical system. These applications are mainly to the dynamics of surface homeomorphisms, where the geometry and algebra involved are both accessible.

Zdeněk Režný, Evžen Kindler (1973)

Aplikace matematiky

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Juan González-Meneses, Bert Wiest (2004)

Annales scientifiques de l'École Normale Supérieure

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Karen Brucks, R. Galeeva, P. Mumbrú, D. Rockmore, Charles Tresser (1997)

Fundamenta Mathematicae

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We discuss a generalization of the *-product in kneading theory to maps with an arbitrary finite number of turning points. This is based on an investigation of the factorization of permutations into products of permutations with some special properties relevant for dynamics on the unit interval.

Roman Srzednicki (1990)

Fundamenta Mathematicae

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Alan R. Camina (1979)

Mathematische Zeitschrift

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Sayed, Mohamed (2005)

International Journal of Mathematics and Mathematical Sciences

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Roman Srzednicki (2015)

Annales Mathematicae Silesianae

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We announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.