On $C^0$ genericity of various shadowing properties
Piotr Kościelniak Marcin Mazur
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 523-530 doi: 10.3934/dcds.2005.12.523
We prove that the following properties are $C^0$ generic in the space of discrete dynamical systems on a compact smooth manifold $M$: periodic shadowing (Theorem 1.1) and, assuming dim$M\leq 3$, $\mathcal T_C$-inverse shadowing (Theorem 1.2).
keywords: inverse) shadowing (discrete) dynamical system covering relations handle decomposition (periodic Generic property
Shadowing is generic---a continuous map case
Piotr Kościelniak Marcin Mazur Piotr Oprocha Paweł Pilarczyk
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3591-3609 doi: 10.3934/dcds.2014.34.3591
We prove that shadowing (the pseudo-orbit tracing property), periodic shadowing (tracing periodic pseudo-orbits with periodic real trajectories), and inverse shadowing with respect to certain families of methods (tracing all real orbits of the system with pseudo-orbits generated by arbitrary methods from these families) are all generic in the class of continuous maps and in the class of continuous onto maps on compact topological manifolds (with or without boundary) that admit a decomposition (including triangulable manifolds and manifolds with handlebody).
keywords: inverse shadowing semidynamical system Shadowing manifold pseudotrajectory periodic shadowing $C^{0}$ topology. continuous surjection genericity continuous map
Semi-hyperbolicity and hyperbolicity
Marcin Mazur Jacek Tabor Piotr Kościelniak
Discrete & Continuous Dynamical Systems - A 2008, 20(4): 1029-1038 doi: 10.3934/dcds.2008.20.1029
We prove that for $\mathcal{C}^1$-diffeomorfisms semi-hyperbolicity of an invariant set implies its hyperbolicity. Moreover, we provide some exact estimations of hyperbolicity constants by semi-hyperbolicity ones, which can be useful in strict numerical computations.
keywords: numerical computations. Hyperbolicity semi-hyperbolicity

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