Diagonal limits of cubes over a finite alphabet

Transcription

1 Diagonal limits of cubes over a finite alphabet Bogdana Oliynyk National University of Kyiv-Mohyla Academy, Kyiv, Ukraine Pilsen, July 2, 2018

2 Hamming space Let B = {b 1,..., b q } be an alphabet, q 2. Denote by H n (q) the Hamming space of dimension n on alphabet B. This space consists of all n-tuples (a 1,..., a n ), a i B, 1 i n, where the distance d Hn between two n-tuples is equal to the number of coordinates where they differ. The scaled Hamming space Ĥn(q) have the same set of points, but the distance is defined as 1 n d H n. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

3 Divisible sequence A sequence of positive integers τ = (m 1, m 2,...) is called divisible if m i m i+1 for all i N. Denote by (s 1, s 2,...) the sequence of ratios of the sequence τ, i.e. s 1 = m 1, s i+1 = m i+1 m i, i 1. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

4 For any i 1 define an isometric embedding ψ si : Ĥm i (q) Ĥm i+1 (q) by the rule: ψ si (x 1,..., x mi ) = (x 1,..., x mi x 1,..., x mi... x 1,..., x }{{ mi ). } s i m i Then the sequence τ determines the directed system of scaled Hamming spaces on the alphabet B Ĥm i (q), ψ si i N with the diagonal embeddings ψ si, i 1. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

5 The limit space of the directed system H(τ, q) = lim Ĥm i (q), ψ si is called a diagonal limit of the sequence of spaces Ĥm i, i 1. We call the metric space (H(τ, q)) the τ-periodic Hamming space over the alphabet B. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

6 Limit of Hamming spaces In [1] Peter J. Cameron and Sam Tarzi considered H(τ, 2) over alphabet {0, 1}. (A) H(τ, 2) isometric to the space of finite unions of half-open subintervals of the interval [0, 1) with some special rational endpoints; (B) Completions of H(τ, 2) independent of choice of τ; (C) For any τ the space H(τ, 2) is homogeneous. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

7 Limit of Hamming spaces In [2] periodic Hamming spaces H(τ, 2) over alphabet {0, 1} were regarded as spaces of clopen subsets of boundaries of spherically homogeneous rooted trees. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

8 Questions In [1] P. J. Cameron and S. Tarzi formulated the questions: (A) Is there an other representation of H(τ, q), q > 2? (B) Are completions of diagonal limits Hamming spaces H(τ, q) on alphabet B independent of choice of τ? Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

9 Rooted tree Let T be an infinite locally finite spherically homogeneous rooted tree with the root v 0, n be some nonnegative integer. A spherically homogeneous rooted tree T is uniquely defined by its spherical index, i.e. by an infinite sequence of positive integers [s 1 ; s 2 ;..., ] such that s i is the number of edges joining a vertex of the (i 1)th level with vertices of the ith level, i 1. If the tree (T, v 0 ) has spherical index [s 1 ; s 2 ;..., ] then the sequence L i = m i, i 1, is divisible. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

10 Boundary of tree The boundary T of a tree T is the set of infinite rooted paths. Define a distance ρ on the set T as { 1 k+1 ρ(γ 1, γ 2 ) =, if γ 1 γ 2, 0, if γ 1 = γ 2 where k is the length of the common beginning of rooted paths γ 1 and γ 2. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

11 Bernoulli measure Define the Bernoulli measure µ on the Borel σ-algebra of T τ by the rule: µ(c v ) = 1 n v, where n v is the number of vertices of T τ on the level containing the vertex v, C v is cylindrical set corresponding to v (see [3]). Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

12 Introduce the discrete metric ϱ on the alphabet B, i.e. { 1, if i j ϱ(b i, b j ) = 0, if i = j, for all 1 i, j q. The metric ϱ induces the discrete topology on B. Denote by C( T τ, B) the set of all continuous functions from the space T τ to the space B. Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

13 Define a metric d µ : C( T τ, B) C( T τ, B) R + by putting d µ (f, g) = µ(σ f Σ g ), where f, g C( T τ, B) and the symmetric difference of Σ f = {f 1 (b i ), 1 i q} and Σ g = {g 1 (b i ), 1 i q} defined by the rule: Σ f Σ g = i j(f 1 (b i ) g 1 (b j )). Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

14 Theorem [4] The τ-periodic Hamming space H(τ, q) on the alphabet B is isometric to the space of all continuous functions C( T τ, B) with the metric d µ. This theorem is the answer to the question (A). Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

15 The following statement is the answer to the question (B). Corollary [4] For every infinite strictly increasing divisible sequence τ = (m 1, m 2,...) the completion of the space H(τ, q) is isometric to the completion of the space H(β, q), where β = (2, 4, 8,...). Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

16 References P. J. Cameron, S. Tarzi, Limits of cubes, Topology and its Applications, V. 155, Is. 14, 2008, pp B. V. Oliynyk, V. I. Sushchanskii The isometry groups of Hamming spaces of periodic sequences, Siberian Mathematical Journal, Vol. 54, N. 1, 2013, pp R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskii, Automata, dynamical systems, and groups (Russian), Proc. Steklov Inst. Math., V. 231, 2000, pp B. Oliynik, The diagonal limits of Hamming spaces, Algebra and Discrete Math.,Vol. 15, N. 2, 2013, pp Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17

17 Thank you for your attention! Bogdana Oliynyk (NaUKMA) Diagonal limits of cubes over a finite alphabet July 2, / 17