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A Cantor dynamical system is slow if and only if all its finite orbits are attracting

  • *Corresponding author: Piotr Oprocha

    *Corresponding author: Piotr Oprocha

This work was supported by National Science Centre, Poland (NCN), grant no. 2019/35/B/ST1/02239

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  • In this paper we completely solve the problem of when a Cantor dynamical system $ (X, f) $ can be embedded in $ \mathbb{R} $ with vanishing derivative. For this purpose we construct a refining sequence of marked clopen partitions of $ X $ which is adapted to a dynamical system of this kind. It turns out that there is a huge class of such systems.

    Mathematics Subject Classification: Primary: 37B10, 37E05; Secondary: 05C38.

    Citation:

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  • Figure 1.  In this representation, the two periodic orbits of the systems are not distinguished in the first graph

    Figure 2.  Illustration of the graph $ G(\mathcal{U}) $ for a supercyclical partition $ \mathcal{U} $ of $ (X, f) $. The dashed regions correspond to the attracted part of the partition, the remainder corresponds to the supercyclical part

    Figure 3.  Removing the divergent vertices in the attracted part

    Figure 4.  Two finite directed graphs $ G = (V, E) $ (left) and $ G' = (V', E') $ (right) and $ \pi: V\rightarrow V' $ morphism, sending vertices of $ G $ to the one of the same color in $ G' $. The graph $ G' $ can correspond to a well-marked partition $ (\mathcal{V}, \tau, \chi) $. However it is not possible to find a partition whose graph is $ G $ and which would be well-marked relatively to $ (\mathcal{V}, \tau, \chi) $. Indeed, whatever the way we mark the red vertices, there will be at least one circuit left with no marker or no potential

    Figure 5.  Illustration on an example of the definition of the graphs $ \mathcal{I}(G, \chi) $ and $ \mathcal{A}(G, \chi) $ for the graph $ G = G(\mathcal{U}) $ and $ \chi: \mathcal{U} \rightarrow \{0, \downarrow, \uparrow, *\} $ where $ (\mathcal{U}, \chi) $ is a well-marked partition; the function $ \chi $ is partially represented (for simplicity): only markers and vertices with $ \chi(u) = 0 $ (the ones in dashed regions) are represented

    Figure 6.  Illustration of the definition of $ \iota_n $ on preimages of some vertex in $ G_{n-1} $ when this vertex is not in a circuit corresponding to a finite orbit

    Figure 7.  Illustration of the definition of $ \iota_n(v) $ for $ v $ a preimage by $ \pi_{n-1} $ of $ w $ which is in a circuit of the attracted part of $ \mathcal{U}_{n-1} $

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