/ 23 Boolen Liner Dynmicl System (Topologicl Mrkov Chin) Yokun Wu (~ˆn) Shnghi Jio Tong University (þ ÏŒA) S ŒAŽA AAH 7 ÊcÊ ÔF(.õ&F)
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph, representing, sy, nonprmetric Mrkov chin
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3} {4}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3} {4} {, 5}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3} {4} {, 5} {2, 3}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3} {4} {, 5} {2, 3} {3, 4}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3} {4} {, 5} {2, 3} {3, 4} {, 4, 5}
Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph nd its evolving vertex suset {2, 5} {3} {4} {, 5} {2, 3} {3, 4} {, 4, 5} {, 2, 3, 5}
Phse spce (ƒ W) 3 / 23 {4, 5} {, 2, 3} {2, 3, 4, 5} {, 3} {2, 4} {, 3, 5} {2, 3, 4} {, 3, 4, 5} {, 2, 3, 4, 5} {, 2, 3, 4} {5} {, 2} {, 2, 4} {, 2, 3, 5} {, 4} {2, 5} {3} {4} {, 5} {2, 3} {3, 4} {, 4, 5} {} {2} {, 2, 5} {3, 5} Figure: All vertices of PS Γ other thn will eventully visit V(Γ), showing tht Γ is primitive
Phse spce (ƒ W) 3 / 23 {4, 5} {, 2, 3} {2, 3, 4, 5} {, 3} {2, 4} {, 3, 5} {2, 3, 4} {, 3, 4, 5} {, 2, 3, 4, 5} {, 2, 3, 4} {5} {, 2} {, 2, 4} {, 2, 3, 5} {, 4} {2, 5} {3} {4} {, 5} {2, 3} {3, 4} {, 4, 5} {} {2} {, 2, 5} {3, 5} Figure: In seven steps {2, 5} moves to set which properly contins itself
Wielndt-type mtrices (Y A.Ý4) 4 / 23 Tke positive integer k 2 nd choose i [k ] stisfying gcd(i, k) =. A Wielndt-type mtrix/digrph W k;i is the mtrix/digrph with vertex set Z/kZ nd rc set {i i + : i =,..., k} {k + i}. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W 4; =, W 0 0 0 4;3 =, W 0 0 0 5;4 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
PS W4; 2 3 4 4 2 23 34 34 24 3 24 23 234 234 In 0 = (4 ) 2 + steps every vertex other thn reches 234. Let the digrph Γ hve k vertices. Wielndt (959) shows tht if every vertex in the phse spce PS(Γ), excepting the vertex, will go to the vertex V(Γ), then every such vertex cn chieve it in t most (k ) 2 + steps nd the ound is ttined if nd only if the digrph Γ is isomorphic to W k;. H. Wielndt, Unzerlegre, nicht negtive Mtrizen, Mthemtische Zeitschrift 52 (959) 642 648. 5 / 23
6 / 23 PS W4;3 2 3 4 2 23 34 3 24 23 234 4 34 24 234 In 6 = 2 4 2 steps every vertex other thn reches 234.
Theorem (W., Zeying Xu, Yinfeng Zhu) Let Γ e strongly connected digrph, nd let d e its dimeter. If S 0 cn rech S in PS Γ, then it cn do tht in t most d S steps. When S is singleton set, this is result of Coxson-Lroson-Schneider (987). Theorem (W., Zeying Xu, Yinfeng Zhu) Let Γ e primitive digrph with girth g nd k vertices. If S V(Γ) is the endpoint of length-i pth in PS Γ, then S i+ k g +. If Γ is the Wielndt digrph W k,, ll equlities in the ove two theorems hold. Wielndt s ound on primitive index nd Akelek-Kirklnd ound (2009) on scrmling index (plus some results on generlized competition index) cn e esily otined from the ove results. 7 / 23
From one to severl MžØW i² r µçcõ Ergodicity coefficients nd John Hjnl 29 Discussions of Mrkov chins... re generlly restricted to the homogeneous cse... In prticulr, the ergodic properties hve only een estlished in this cse. It seems nturl to consider to wht extent these properties hold for non-homogeneous chins. J. Hjnl, (956). The ergodic properties of non-homogeneous finite Mrkov chins. Proc. Cmridge Philosoph. Soc. 52:67õ77. Downloded y [Shnghi Jio Tong University School of Medicine] t 8:48 7 Mrch 205 Figure. John Hjnl (924 2008). John Hjnl ws orn in Drmstdt, Germny, on Novemer 26, 924 Kálmn nd Ev Hjnl-Kónyi. His Jewish fmily hd strong Hungrin identit nd From: pssion for Germn E. high Senet, culture. John ws (204). intellectully gifted nd linguis With the onset of the Nzi regime, while his prents prepred to leve Germny fo Englnd Inhomogeneous in 936, John ws sent to Quker school Mrkov in the Netherlnds. chins After ye he ws reunited with his prents in London efore the Netherlnds were overrun the Nzis. Although nd his ergodicity youthful interests werecoefficients: in science nd mthemtics, he missed o on forml mthemticl eduction, grduting from Oxford in 943 with First Cl Honors John in Politics, Hjnl Philosophy, nd (924õ2008). Economics. He ws t the Royl Commissio on Popultion, 944 948. On the strength of his work, he ws recruited to wor on demogrphy Communictions the UN in New York. In 95, in he Sttistics left this prestigious nd we pid jo to go to the Office of Popultion Reserch t Princeton University. Tkin courses in mthemtics, he cme in contct with Willim Feller whose celerte tretise, Theory An Introductionnd to Proility Methods. Theory nd Its Applictions, vol. 43. hd een fir pulished in 950. Hjnl s very first pper in 956 (on strong ergodicity), 8 / 23 gives h
Dynmicl system sed on severl digrphs 9 / 23 They re studied under different nmes, sy inhomogeneous Mrkov chin, switched liner system, synchronous Boolen network, utomt theory, symolic dynmics,... Some typicl prolems: rechility, controllility, oservility, existence of decomposition into severl digrphs with prescried dynmicl properties,...
Four digrphs on the sme vertex set 0 / 23 2 3 2 3 4 4 2 3 2 3 4 4
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4} {, 3, 4}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4} {, 3, 4} {3, 4}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4} {, 3, 4} {3, 4} {, 2, 4}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4} {, 3, 4} {3, 4} {, 2, 4} {2, 4}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4} {, 3, 4} {3, 4} {, 2, 4} {2, 4} {, 4}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 3, 4} {2, 3, 4} {, 3, 4} {3, 4} {, 2, 4} {2, 4} {, 4} {4}
Phse trnsitions (ƒ) / 23 2 3 4 {2, 3, 4} {, 3, 4} {3, 4} {, 2, 4} {2, 4} {, 4} {4} {, 2, 3}
Phse trnsitions (ƒ) / 23 2 3 4 {, 3, 4} {3, 4} {, 2, 4} {2, 4} {, 4} {4} {, 2, 3} {2, 3}
Phse trnsitions (ƒ) / 23 2 3 4 {3, 4} {, 2, 4} {2, 4} {, 4} {4} {, 2, 3} {2, 3} {, 3}
Phse trnsitions (ƒ) / 23 2 3 4 {, 2, 4} {2, 4} {, 4} {4} {, 2, 3} {2, 3} {, 3} {3}
Phse trnsitions (ƒ) / 23 2 3 4 {2, 4} {, 4} {4} {, 2, 3} {2, 3} {, 3} {3} {, 2}
Phse trnsitions (ƒ) / 23 2 3 4 {, 4} {4} {, 2, 3} {2, 3} {, 3} {3} {, 2} {2}
Phse trnsitions (ƒ) / 23 2 3 4 {4} {, 2, 3} {2, 3} {, 3} {3} {, 2} {2} {}
The output ( Ñ) / 23 2 3 4 {4} {, 2, 3} {2, 3} {, 3} {3} {, 2} {2} {}
Phse spce (ƒ W) 2 / 23 {, {, {, 2, {, 2, 3, 2, 3, 2, 4} 3, 4} 3, 4} 4} {2, 3, 4} {, 3, 4} {3, 4} {, 2, 4} {2, 4} {, 4} {4} {, 2, 3} {2, 3} {, 3} {3} {} {2} {, 2}
Annihilting pth ("z ) 3 / 23 Let Γ e set of digrphs on the sme vertex set [k]. A pth from [k] to in PS Γ, if ny, is clled n nnihilting pth. Wht is the length of shortest nnihilting pth? Sntesso nd Vlcher (2008) estlish n esy upper ound 2 k nd they construct digrph set of size 2 k 2 to show tht the ound is tight. Theorem (W., Yinfeng Zhu) There exists set Γ of k digrphs on the sme vertex set [k] such tht PS Γ contins unique nnihilting pth of length 2 k.
Primitive digrph set (k 68Ü) 4 / 23 Let Γ e set of digrphs on the sme vertex set [k]. We cll Γ primitive provided every "long" wlk in PS Γ strting from nonempty suset S of [k] will lwys hit [k] (nd then sty there forever). If Γ is primitive, the length of longest pth in PS Γ is clled the primitive index of Γ, denoted g(γ).
Primitive digrph set (k 68Ü) 5 / 23 F 3 2 3 2
Primitive digrph set (k 68Ü) 5 / 23 F 3 2 3 2 PS(F ) {} {2} {3} {, 3} {, 2} {2, 3} {, 2, 3} In 2 3 2 = 6 steps we ll rrive t {, 2, 3}.
6 / 23 Let α k denote the mximize size of primitive digrph set on the vertex set [k]. For t α k, let g k,t e the lrgest primitive index of size-t primitive digrph set on vertex set [k]. Theorem (W., Yinfeng Zhu) g k, g k,αk. Wielndt s Theorem sys tht g k, = (k ) 2 +. Cohen nd Sellers find tht g k,2 k 2 = 2 k 2. They define γ k to e the minimum t such tht g k,t = 2 k 2. Theorem (W., Yinfeng Zhu) γ k k. It is known tht γ 2 =, γ 3 = 2, γ 4 = 3, γ 5 {2, 3, 4}. We even do not know if it holds γ 4 γ 5 γ 6.
7 / 23 Let k e n integer not less thn 2 nd M primitive set of digrphs on the vertex set [k]. Let γ(m) e the minimum size of digrph set F such tht g(f M) = 2 k 2. Theorem (W., Yinfeng Zhu) γ(m) 2 k 3. Theorem (Xinmo Wng, W., Ziqing Xing) γ(w k,k ) ( (k 2)/2 ) for ll integers k 2. The proof of the ound for γ(w k,k ) mkes use of pir of symmetric chin decompositions of the Boolen lger.
8 / 23 Let Γ e primitive set of digrphs on the vertex set [k]. Let r = Γ. Question (W., Yinfeng Zhu) If we know the existence of pth in PS Γ from S 0 to S, how to otin n upper ound for its length in terms of the size of the trget set S nd the prmeter r? Wht out S k r? Question (W., Yinfeng Zhu) Wht is the length of longest pth in PS Γ in which every vertex is suset of [k ]? Is this length ounded y k r?
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4} {, 2, 4}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4} {, 2, 4} {, 2}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4} {, 2, 4} {, 2} {2, 3}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4} {, 2, 4} {, 2} {2, 3} {3, 4}
Synchronizing word (ÓÚzi) 9 / 23 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4} {, 2, 4} {, 2} {2, 3} {3, 4} {, 4}
Synchronizing word (ÓÚzi) 4 3 2 {, 2, 3, 4} {, 2, 3} {2, 3, 4} {, 3, 4} {, 2, 4} {, 2} {2, 3} {3, 4} {, 4} {} The length of the synchronizing word is (4 ) 2 = 9. Černý Conjecture (964): If Γ consists of set of constnt outdegree- digrphs on vertex set [k] nd if there is pth in PS Γ contining oth [k] nd singleton set, then such pth cn e chosen to e no longer thn (k ) 2. 9 / 23
20 / 23 Removing the constnt outdegree- condition Upper ound for the length of shortest wlk including [k] nd one singleton set? Upper ound for the length of shortest wlk including [k] nd two singleton sets?... Upper ound for the length of shortest wlk including [k] nd k singleton sets: 2 k 2.
Grph-indexed Mrkov chin (6œ«2¼Q) 2 / 23 If we think of time not s infinite pth (integers) ut s generl grph, we go from usul Mrkov chin to grph-indexed Mrkov chin. Tke grph G. A function f from V(G) to integers is (α, β)-lipschitz if min f = 0 nd α dist G (u, v) f (u) f (v) β dist G (u, v) for ll u, v V(G). The height of such mp f is mx f. The verge height of ll (α, β)-lipschitz functions on G is clled the (α, β)-height of G.
Grph-indexed Mrkov chin: Contd. 22 / 23 Theorem (W., Zeying Xu, Yinfeng Zhu) Among ll trees with the sme numer of vertices, the pth is the unique tree whose (α, β)-height tkes the mximum vlue (provided α < β nd there is n (α, β)-lipschitz function on the pth). Conjecture (Benjmini-Schechtmn, Loel-Nešetřil-Reed) The pth is indeed the extreml grph mong ll grphs with the sme numer of vertices.
23 / 23 My techer: Jiong-Sheng Li (oá)) 935 205 My couthors: Acknowledgement ( N) Yinfeng Zhu (6Ç ), Mster student, Shnghi Jio Tong University Zeing Xu (MP ), Ph.D student, Shnghi Jio Tong University Xinmo Wng (#j), University of Science nd Technology of Chin Ziqing Xing ( f ), Ph.D student, University of Georgi Thnk you!