On minimal maps of 2-manifolds

Blokh, Alexander; Oversteegen, Lex G.; Tymchatyn, E. D.

doi:10.1017/s0143385704000331

Cited by 16 publications

(24 citation statements)

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“…It is well known that a minimal system contains neither subinvariant nor superinvariant (i.e., such that T (A) ⊂ A or T (A) ⊃ A, respectively) nonempty proper closed subsets (see [5,Lemma 3.10]). It is also well known that if X is connected then every minimal map T is totally minimal, i.e., all iterates T n , except of identity, are minimal.…”

Section: Preliminariesmentioning

confidence: 99%

“…This implies that the complexity c(n) of this subshift (the number of blocks of length n) satisfies c(n) ≥ n + 1 (see [17]). 5 Let be the smallest distance between the atoms of P. The complexity yields that given n there exist (more than) n (n, )-separated points for T . Further, there exist (more than) n!…”

Section: Theorem 2 Let T : X → X Be a Self-homeomorphism Of A Compactmentioning

confidence: 99%

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Minimal Spaces with Cyclic Group of Homeomorphisms

2015

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There are two main subjects in this paper.(1) For a topological dynamical system (X, T ) we study the topological entropy of its "functional envelopes" (the action of T by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of X ). In particular we prove that for zero-dimensional spaces X both entropies are infinite except when T is equicontinuous (then both equal zero). (2) We call Slovak space any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems (X, T ) with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.

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Section: Preliminariesmentioning

confidence: 99%

Section: Theorem 2 Let T : X → X Be a Self-homeomorphism Of A Compactmentioning

confidence: 99%

Minimal Spaces with Cyclic Group of Homeomorphisms

2015

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“…A natural question then is whether this is a general property of minimal maps on the 2-torus, or more generally, on any 2-dimensional manifold. As it turns out, the Main Theorem together with some dynamical tools allow us to prove the following theorem which will appear in [4]. We hope that the Main Theorem can be helpful in studying minimal maps of manifolds of higher dimension.…”

Section: Main Theorem Suppose That F : M → N Is a Light And Almostmentioning

confidence: 95%

“…Another motivation which we had was that this result might be useful in studying minimal maps. Indeed, it has been instrumental in obtaining new results for minimal maps and related questions (see [4]). Hence our interest in almost 1-to-1 maps is explained by the fact that they are natural candidates for being homeomorphisms/embeddings, but also by the fact that all minimal maps are almost 1-to-1 ( [7]).…”

Section: Introductionmentioning

confidence: 99%

On almost one-to-one maps

Blokh

Oversteegen

Tymchatyn

2006

Trans. Amer. Math. Soc.

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Abstract. A continuous map f : X → Y of topological spaces X, Y is said to be almost 1-to-1 if the set of the points x ∈ X such that f −1 (f (x)) = {x} is dense in X; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and σ-compact spaces (e.g., n-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if f is a minimal self-mapping of a 2-manifold M , then point preimages under f are tree-like continua and either M is a union of 2-tori, or M is a union of Klein bottles permuted by f .

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“…Note that, if M is a compact connected two-dimensional manifold and f : M → M is a continuous map and K ⊆ M is a minimal set of (M, f ) then either K = M or K is a nowhere dense subset of M . Moreover, the former case is possible only if M is a torus or a Klein bottle [Blokh et al, 2005]. On manifolds of dimension greater than two, a general theorem by Katok [1972] and Fathi and Herman [1977] ties the existence of minimal diffeomorphisms to the existence of locally free diffeomorphisms.…”

Section: Introductionmentioning

confidence: 99%