**Title:** Lipschitz and Holder shadowing property and Inhomogeneous linear equation

**Speaker:** Sergey Tikhomirov

**Speaker Info:** Freie Universität Berlin

**Brief Description:**

**Special Note**:

**Abstract:**

The shadowing problem is related to the following question: under which condition, for any pseudotrajectory (approximate trajectory) of a vector field there exists a close trajectory? We proved that Lipschitz shadowing is equivalent to hyperbolicity (structural stability).The main technique is consideration of bounded solutions of inhomogeneous linear equation $$ v_{k+1} = A_k v_k + w_{k+1}, $$ where $A_k$ are differential of the diffeomorphism along an exact trajectory and $w_k$ is an arbitrarily bounded sequence.

Those results can be generalized to Holder shadowing on finite intervals. In that case notion of bounded solution is replaced by solutions with sublinear growth.

We discuss connections of this problem with Katok's question: "Does any diffeomorphism Holder conjugated to Anosov must be Anosov by itself?" and Hammel-Grebogi-Yorke conjecture on shadowability of Henon map.

Talk in based on joint results with Prof. Pilyugin.

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