Transitive set

Transitive set

Dynamical Systems and Linear Algebra

Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006

A set ℳ ⊂ M is a chain recurrent component, if it is a maximal (with respect to set inclusion) chain transitive set. In this case ℳ is a connected component of the chain recurrent set Cℛ.

On the computation of extended rescaled Poincaré maps for singular vector fields

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Published in Dynamical Systems, 2020

Qianying Xiao, Yiwei Zhang

For any , a collection is called a ε -pseudo-orbit for the flow , if for any , A point x is chain recurrent if there exist pseudo-orbits from x to x with arbitrary small jumps. A compact invariant set Λ is aperiodic chain transitive if Λ contains no periodic points, (i.e. for any point and t>0, ), and for any two points , there exist pseudo-orbits in Λ from x to y with arbitrary small jumps. A chain recurrent class is defined as a maximal chain transitive set.

Expansive transitive sets for robust and generic diffeomorphisms

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Published in Dynamical Systems, 2018

Manseob Lee, Junmi Park

Let and Λ is a locally maximal transitive set. Suppose that f|Λ is expansive. Then, there exists a constant C > 0, 0 < λ < 1 and such that for each q which is a periodic point of f in Λ with period π(q, g) ≥ m, where k = [π(q, g)/m].Λ admits a dominated splitting TΛM = E⊕F with dimE = index(p).

Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

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Published in Dynamical Systems, 2022

Dawei Yang, Jinhua Zhang

There is a dense set in such that for any , if Λ is a non-trivial isolated transitive set, then Λ is a local homoclinic class .

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