Structure theorem on random permutations Jisang Yoo Sungkyunkwan - PowerPoint PPT Presentation

Structure theorem on random permutations Jisang Yoo Sungkyunkwan University August 23th, 2019 Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 1 / 12

random permutation Objects in this talk are stationary sequences of random permutations. Most results are implied by the following two papers independently. M. Klunger, Periodicity and Sharkovsky’s theorem for random dynamical systems. Stochastics Dynamics 1 (2001) 299-338. J. Yoo, Multiplicity structure of preimages of invariant measures under finite-to-one factor maps. Trans. Amer. Math. Soc. 370 (2018) Overlap between theory of random dynamical systems and theory of factor maps (between dynamical systems). Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 2 / 12

Definition Let X := { 1 , 2 , · · · , d } (a fixed finite set of d symbols) S d := the symmetric group of degree d or { permutations on X } To have a random permutation, we need a probability space (Ω , P ): A random permutation is just a measurable function : F : Ω → S d , ω �→ F ω To have a sequence of random permutations, we need a transformation θ : Ω → Ω that is measure preserving, (wlog) invertible, ergodic. We’ll call the pair ( θ, F ) a random permutation system. It’s the same thing as stationary sequences of random permutations because ω �→ ( F ω , F θ ( ω ) , F θ 2 ( ω ) , · · · ) does form a stationary sequence, and any stationary sequence can be realized in this form. There are two ways of visualizing this. Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 3 / 12

What can we ask If we fixed the initial point x 0 ∈ X , what is the long term behavior of the random orbit : x 0 , F ω ( x 0 ) , F θ ( ω ) ( F ω ( x 0 )) , · · · Each initial point x 0 ∈ X and noise ω ∈ Ω determines a unique orbit ( x 0 , x 1 , x 2 , · · · ) ∈ X N . Does the orbit visit a given point x ′ ∈ X with a fixed frequency? If so, when does the frequency depend only on x ′ and when does it depend on both x ′ , x 0 ? Given ω ∈ Ω, it does not determine the orbit ( x 0 , x 1 , x 2 , · · · ) uniquely, but it gives exactly d possible orbits depending on x 0 . In particular, the orbit ( x 0 , x 1 , x 2 , · · · ) is not a stationary sequence of random points. So the ergodic theorem does not directly apply to orbits. Let’s answer these questions from a structure theorem. Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 4 / 12

Remarks Random permutation systems generalize to: random dynamical systems (replace permutations with homeomorphisms) random subshift of finite type (replace permutation matrices with zero-one matrices) (Y. Kifer’s definition, not K. McGoff) matrix cocycle (replace permutation matrices with arbitrary matrices. Objects of multiplicative ergodic theorem) factor map (to be explained) All the more reasons for existence of other papers implying the results here. If there is, let me know. Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 5 / 12

Structure result: first part Let ( θ, F ) be a random permutation system. Recall θ : Ω → Ω is a measure preserving transformation of (Ω , P ). Recall that over a.e. ω ∈ Ω, there is a permutation F ω : X → X , where X is a finite set of d symbols. Then, over a.e. ω ∈ Ω, we can associate a partition of the state space X = { 1 , 2 , · · · , d } into d ′ sets, d ′ ≤ d : { 1 , 2 , · · · , d } = ∪ d ′ i =1 Q i ( ω ) such that for each 1 ≤ i ≤ d ′ , the permutations F ( ω ) , F ( θω ) , F ( θ 2 ω ) , · · · restricts to Q i ( ω ) , Q i ( θω ) , Q i ( θ 2 ω ) , · · · compatibly and results in another sequence of random permutations on a smaller number of symbols (caution!). The partition can be chosen to be maximal (and measurably). This maximal invariant partition is unique. To have a better picture, with abuse of notation, let Q i := ∪ ω ∈ Ω { ω } × Q i ( ω ) ⊂ Ω × X . Then Q i , 1 ≤ i ≤ d ′ , form a maximal invariant partition of Ω × X . Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 6 / 12

second part If d ′ = 1, we’ll say ( θ, F ) is minimal. For any minimal random permutation system, for a.e. ω ∈ Ω, the d orbits visit a given state x ′ ∈ X with same frequency, namely, 1 d . In particular, each orbit visits everywhere. The d orbits visit a given finite sequence of states with same frequency. For a non-minimal random permutation system time means from the d orbits exist, but they may be different among the d orbits. Nonetheless, there can be only up to d different time means. The set of possible time means do not depend on ω . The number of time means ≤ d ′ ≤ d . To have a better picture, each Q i is a minimal permutation system on its own, and each of these is ( P -relatively) uniquely ergodic. Everything else follows from this property. Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 7 / 12

What are examples of minimal random permutations? For this slide, assume Ω = S Z d and θ : Ω → Ω is the shift, and F ω := ω 0 . The following cases are all minimal: the i.i.d. sequence of permutations: i.e. P := a product measure. (module some obvious obstruction) P := (irreducible, stationary) Markov chain, regular equilibrium state (implied by Tuncel 1981, Walters 1986) Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 8 / 12

Examples of non-minimal Just take a direct sum of two random permutation systems. Each orbit visiting everywhere does not imply minimality. For a counter-example, take the above example and apply cohomologous change. Each orbit visiting everywhere and being natural does not imply minimality. For a counter-example, try X = { 1 , 2 } and for a.e. ω , make one orbit be the ( 1 3 , 2 3 )-Bernoulli process. More examples in the second mentioned paper. Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 9 / 12

random permutations as factor maps For this slide, assume Ω = S Z d and θ : Ω → Ω is the shift, and F ω := ω 0 . Then θ : Ω → Ω is the full shift with alphabet S d . The transformation on Ω × X is an extension of Ω and is conjugate to a subshift of finite type inside Ω × X Z . This induces a finite-to-one factor map from the SFT to the full shift. And converse. Given a finite-to-one factor map from an SFT X ′ to its image Y ′ (not necessarily full shift) and an ergodic measure ν on Y ′ , it induces a random permutation system. Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 10 / 12

Applications? In the satellite conference, I will discuss some application to a special class of random dynamical systems. Please find other applications in or outside dynamics. If you are working on a problem and you see finite bundles and dynamics, random permutation theory applies. For each random permutation system, you can ask: is it minimal? If not, how many minimal components? What is the size of partition elements? Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 11 / 12

Thanks Thank you! Jisang Yoo (SKKU) Structure theorem on random permutations August 23th, 2019 12 / 12