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The perturbation to the Noether symmetry and the Noether adiabatic invariants of discrete difference variational Hamilton systems are investigated. The discrete the Noether exact invariant induced directly by the the Noether symmetry of the system without perturbation is given. The concept of discrete high-order adiabatic invariant is presented, the criterion of the perturbation to the Noether symmetry is established, and the discrete the Noether adiabatic invariant induced directly by the perturbation to the Noether symmetry is obtained. Lastly, an example is discussed to illustrate the application of the results.

Perturbation to Noether Symmetry and Noether Adiabatic Invariants of Discrete Difference Variational Hamilton Systems