Plaque expansivity conjecture for partially hyperbolic diffeomorphisms

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Let ff be a partially hyperbolic, dynamically coherent diffeomorphism. It is plaque expansive if there exists ε>0\varepsilon>0 such that any two ε\varepsilon-central pseudotrajectories yk\\{y_k\\} and zk\\{z_k\\} satisfying

dist⁡(yk,zk)<ε,k∈Z,\operatorname{dist}(y_k,z_k)<\varepsilon,\qquad k\in\mathbb Z,

also satisfy

z0∈Wεc(y0).z_0\in W^c_{\varepsilon}(y_0).

Plaque expansivity conjecture. Any partially hyperbolic, dynamically coherent diffeomorphism is plaque expansive.

Plaque expansivity is a uniqueness-type property for central pseudotrajectories and is important in the theory of partially hyperbolic systems. The source presents the assertion as a conjecture and gives no resolution status here.

References

Primary source

Sergey Kryzhevich and Sergey Tikhomirov, “Partial hyperbolicity and central shadowing”, arXiv:1112.4272 (2012).

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