A Note on Double Minimality
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A Note on Double Minimality Wen Huang1 · Xiangdong Ye1
Received: 21 January 2015 / Revised: 1 March 2015 / Accepted: 4 March 2015 / Published online: 22 March 2015 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2015
Abstract It is known that a minimal prime system is either a subshift or with a connected phase space (Keynes and Newton Trans Am Math Soc 217:237–255, 1976). We show that a double minimal system is a subshift; this implies immediately that no non-periodic map has 4-fold topological minimal self-joinings. We also prove that a POD system is either uniformly rigid or is a subshift. Keywords
Double minimal · POD · Uniformly rigid
Mathematics Subject Classification
54H20
1 Introduction By a topological dynamical system (X, T ), we mean that X is a compact metric space and T : X −→ X is a homeomorphism. A class of weakly mixing systems, called n-fold (n ≥ 2) topological minimal selfjoinings, was studied by Junco [7], King [10], Weiss [12], and others. A dynamical system is doubly minimal if it has 2-fold topological minimal self-joinings, that is for all x ∈ X , y ∈ {T n x}n∈Z , {(T i x, T j y)} j∈Z is dense in X × X . The first example of a doubly minimal system was given by King [10] who also proved that no non-
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Xiangdong Ye [email protected] Wen Huang [email protected]
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Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China
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W. Huang, X. Ye
periodic map has 4-fold topological minimal self-joinings. In [12] Weiss showed that any ergodic system with zero entropy has a uniquely ergodic model which is doubly minimal. In this note we show that doubly minimal systems are subshifts. This fact immediately implies that no non-periodic map has 4-fold topological minimal selfjoinings. A minimal system is prime if it only has trivial factors. It is known that a minimal prime system is either a subshift or with connected phase space [9, Proposition 2.11]; and that both situations can occur. Our theorem indicates that as a special class of prime system, the second situation does not occur for doubly minimal systems. A totally minimal homeomorphism T : X −→ X is called proximal orbit dense (POD) if whenever x = y, o(x, y) (the orbit closure of (x, y) under T × T ) contains An = {(z, T n z) : z ∈ X } for some n = 0. It is easy to see that a dynamical system is POD if and only if (X, T ) is totally minimal and whenever x, y ∈ X with x = y, then for some n = 0, (T n y, x) is proximal. It is easy to see that non-periodic doubly minimality ⇒ POD ⇒ primeness. We show that a POD system is either uniformly rigid or is a subshift. In the rest of the section we introduce some necessary notions. Let X be a compact metric space and T : X −→ X be a homeomorphism. For x ∈ X , o(x), o+ (x), o− (x) are the orbit, forward orbit and the backward orbit of x, respectively. One understands ω(x) (t
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