All minimal Cantor systems are slow

Abstract

We show that every (invertible or noninvertible) minimal Cantor system embeds in double-struckR with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

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

“…Purely attracting zero-dimensional systems. This section contains the main changes compared to the initial construction of [3]. In order to prove the Theorem 1.2, we will consider graph coverings representations in the case of zerodimensional systems with only attracting finite orbits, which is more complex than the minimal ones.…”

“…Still, it was not clear how much vanishing derivative is correlated with the existence of a metric making the map an isometry or at least having entropy zero. At this point it was expected that such an embedding may not exist for expansive maps (in particular subshifts) since these systems have divergence of orbits hidden in the dynamics (see discussion in [3]). Then in [3], J.P.Boroński, J.Kupka and P.Oprocha brought a surprising answer to this, showing that every minimal dynamical system on a Cantor set can be embedded in the real line with vanishing derivative everywhere.…”

“…At this point it was expected that such an embedding may not exist for expansive maps (in particular subshifts) since these systems have divergence of orbits hidden in the dynamics (see discussion in [3]). Then in [3], J.P.Boroński, J.Kupka and P.Oprocha brought a surprising answer to this, showing that every minimal dynamical system on a Cantor set can be embedded in the real line with vanishing derivative everywhere. The proof of this result makes use of J.-M.Gambaudo and M.Martens [8] representation of minimal dynamical systems on the Cantor set with graph coverings which satisfy certain properties.…”

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

“…Still, it was not clear how much vanishing of the derivative is correlated with existence of metric making the map isometry or at least entropy zero. At this point it was expected, that such embedding may not exist for expansive maps (in particular subshifts) since these systems have divergence of orbits hidden in the dynamics (see discussion in [3]). Then [3] by J.P.Boroński, J.Kupka and P.Oprocha brought a surprising solution, showing that every minimal dynamical system on a Cantor set can be embedded into the interval with vanishing derivative everywhere.…”

“…At this point it was expected, that such embedding may not exist for expansive maps (in particular subshifts) since these systems have divergence of orbits hidden in the dynamics (see discussion in [3]). Then [3] by J.P.Boroński, J.Kupka and P.Oprocha brought a surprising solution, showing that every minimal dynamical system on a Cantor set can be embedded into the interval with vanishing derivative everywhere. The proof of this result makes use of J.-M.Gambaudo and M.Martens [7] representation of minimal dynamical systems on the Cantor set with graph coverings which satisfy certain properties.…”