m3/4

Prompt

RESEARCH QUESTION (plane topology / dynamics). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference; unchecked?), LAB (from this prompt, status as given), CONJECTURE, OPEN, or RESTATEMENT (a gap reworded). If a definition is ambiguous, state your reading in one line and continue. Say where your argument stops. A. DEFINITIONS A1 X in R^2: a nondegenerate continuum, U = R^2 \ X connected. T(A) = A plus the bounded components of R^2 \ A. win = winding number. A2 h: X -> X is a homeomorphism of X only (not assumed to extend). g is continuum-minimal (cm) if no nonempty proper subcontinuum K has g(K) in K. Setting (L2): h^2 is cm. Then (LAB, audited) h, h^2 have no fixed point, m := min over x of min(|hx-x|,|h^2x-x|) > 0, X is indecomposable, hereditarily unicoherent, int X empty. A3 F: R^2 -> R^2 is a perfect extension of h: F|X = h and F is "a closed continuous function each of whose point inverses is compact" (BFMOT Def 2.3.1). BFMOT = Blokh, Fokkink, Mayer, Oversteegen, Tymchatyn, Fixed point theorems for plane continua..., Mem. AMS 2013 (quoted: arXiv:1004.0214v2). A4 Crosscut: an open arc q in U whose closure is an arc with distinct ends in X. Sh(q) = bounded component of R^2 \ (X u q), Out(q) = the unbounded one. diam q = diam cl q. A5 (BFMOT 3.6) phi: Riemann map, outside of unit disk onto U u {inf}; ray R_t = phi({r e^(2 pi i t): r > 1}). A chain: disjoint crosscuts Q_i, diam -> 0, Q_j in Sh(Q_i) for j > i; equivalent chains form a prime end E_t. Pr(E_t) = {z: a chain defining E_t converges to z}. The (X,x)-end of R_t (x on R_t): the bounded component of R_t \ {x}; R_t crosses Q essentially iff some (X,x)-end lies in Sh(Q). x in X is accessible iff some arc in U u {x} ends at x. A6 Variation (BFMOT Def 2.2.1-2.2.2). A junction J_v: the image of the rays i[0,inf) = J^i, [0,inf) = J^+, -[0,inf) = J^- under an orientation-preserving homeomorphism of C taking 0 to v. For a simple closed curve S, f: S -> C and an arc A = [a,b] in S with f(a), f(b) in T(S), f(A) n A empty: take v in A, J_v with J_v n S = {v}; along M = f^-1(J_v) n [a,b] in counterclockwise order count +1 each time a point of f^-1(J_v^+) is immediately followed by one of f^-1(J_v^i), -1 for the reverse; the sum is var(f,A,S). For a crosscut q, F(cl q) n cl q empty: var(F,q^+) := var(F,q,S), q run counterclockwise in a simple closed curve S with X in T(S) (S-independent: BFMOT Prop 3.4.3); var(F,q^-) := -var(F,q^+). A7 BFMOT standing hypothesis: "f : C -> C is a perfect map, X is a continuum, f has no fixed point in T(X) and X is minimal with respect to f(X) in T(X)". Def 5.1.1: an outchannel is "a prime end E_t ... such that for some chain {g_i} of crosscuts defining E_t, var(f, g_i, T(X)) != 0 for every i" (extraction drops the slash of !=). Thm 6.2.1: "there exists a unique geometric outchannel E_t for X, which is dense in X ... for any sufficiently small chord g in any chain defining E_t, var(f, g, X) = -1, and for any sufficiently small chord g' not crossing R_t essentially, var(f, g', X) = 0." In (L2) it applies to F and to F^2 (LAB, audited): outchannels E_t (ray R_t) of F, E_s (ray R_s) of F^2; Pr(E_t) = X. CP1 (LAB reading, load-bearing, not settled; BFMOT says "chord"): there are delta_t, delta_s > 0 such that every crosscut q with diam q < delta_t has var(F,q^+) = -1 if R_t crosses q essentially, 0 otherwise; the same for F^2, R_s, delta_s. A8 Access (LAB, audited): a closed set rho in U with a proper homeomorphism [0,inf) -> rho; start v = rho(0). For a path beta missing v with ends off rho: I(beta,rho) := win(beta * delta, v), delta any path in R^2 \ rho from beta(1) to beta(0). I is invariant under homotopy rel ends in R^2 \ {v}, so equal for paths with the same ends in a convex open set missing v. For an E-crosscut c: v_c = first point of cl c met by R_t coming from infinity (v_c in c); rho_c = the part of R_t from v_c to infinity. For a path P with ends in X and w in U off P: I(P,w) := I(P,rho_w), any access rho_w from w (independent of rho_w). For a crosscut q with cl q n P empty: I(P,q) := I(P,w), w on q (independent of w). A9 E-crosscut: R_t crosses it essentially; M-crosscut: otherwise. q is far from S if cl q n S is empty. For disjoint E-crosscuts, q is deeper than c iff q in Sh(c). A crosscut d run by a path has sign eps(d) = +1 if run as d^+, else -1; then I(d,w) = eps(d)[w in Sh(d)] for w in U off cl d (LAB, audited). A10 Chain path: a path P with ends in X, each component of P \ X (a piece) a crosscut. E-word: its E-pieces in path order with signs. For an E-crosscut g far from P: D_g(P) := sum of eps(e) over E-pieces e of P with g in Sh(e). Lambda_P(x), x in X off P: the common value of D_g(P) for all deep enough E-crosscuts g near x (exists by Z). Two pieces straddle a far crosscut q if one lies in Sh(q), the other in Out(q). A11 Scales (LAB, audited): r' <= m/6; r_1 <= r' with F(B(y,r_1)) in B(h(y),r') for y in X, 2r_1 < min(delta_t,delta_s); delta_0 <= r_1 with F(B(x,delta_0)) in B(h(x),r_1). Small = diam < delta_0. For small E-crosscut c = (a,b): a first-generation arc is an arc gamma in B(h(a),r_1) from h(a) to h(b) (a chain path); second-generation paths gamma_d run from h(d^-) to h(d^+) in B(h^2(a),r'), one per piece d. V1(c) = var(F,c^+) = -1, V2(c) = var(F^2,c^+) in {0,-1} (CP1). a_c(q) := I(F o c^+, q); it equals I(gamma,q) if q also misses B(h(a),r_1). B. LEMMAS (status; where used) N (PROVED). Disjoint crosscuts c, d: c in Sh(d) or c in Out(d); if c in Sh(d) then Sh(c) in Sh(d); if neither lies in the other's shadow, the shadows are disjoint. Proof: c is connected and misses d. If c in Sh(d) and d in Sh(c), Out(c) u c is connected, unbounded, misses X u d, so c in Out(d): impossible. So d in Out(c), and Sh(c) u c is connected, misses X u d, meets Sh(d). If c in Out(d), d in Out(c) and Sh(c) meets Sh(d), then Sh(c) u c lies in Sh(d): impossible. T1 (PROVED). A disjoint E-crosscut e and M-crosscut m: e in Out(m); disjoint E-crosscuts are nested. Proof: e in Sh(m) gives Sh(e) in Sh(m) (N), so Sh(m) holds an X-end of R_t. Two E-shadows hold nested X-ends of R_t, so they meet; apply N. Used: I, Z, SS. DC (SOURCE BFMOT Lemma 4.4.8 + LAB, audited): each x in X = Pr(E_t) is the limit of a chain of small E-crosscuts (BFMOT KP-chords) defining E_t, their shadows shrinking into X; every E-crosscut starts such a chain. Used: Z, K. L4.3 (LAB, audited). j = 1,2, crosscut c = (a,b), F^j(cl c) n cl c empty, access rho from v in c with rho n (X u cl c) = {v}, B open convex with F^j(cl c) in B, v not in B: var(F^j,c^+) = I(beta,rho) for every path beta in B from h^j(a) to h^j(b). Used: 5.1, D, R. I (PROVED: A9, T1, compactness). For a chain path P and an E-crosscut q far from P: I(P,q) = D_q(P), finitely many terms (rho_w meets finitely many pieces; M-pieces give 0 by T1). So a_c(q) = D_q(gamma) if q misses B(h(a),r_1). Used: RIG, R. 5.1 (LAB, audited). V1(c) = sum over pieces d of gamma of eps(d)[v_c in Sh(d)]; V2(c) = sum over d of I(gamma_d, rho_c) = I(F o gamma, rho_c); finitely many nonzero terms. (L4.3 with rho_c; compactness.) Used: D, D3, R. Z (LAB, UNAUDITED). P a chain path u -> v, d_1, d_2 crosscuts ending at u, v, kappa a closing path (part of d_1 from u, a path in U, part of d_2 to v), Theta = kappa.P (a loop). Some a(kappa) has win(Theta,.) = a(kappa) + D_g(P) on every deep enough E-crosscut g near any x in X off P u cl d_1 u cl d_2. Sketch: win = I(kappa,.) + sum over pieces of I(d,.); M-pieces give 0 (T1); I(kappa,.) is constant on deep shadows, which are nested. Used: SS. SS (LAB, UNAUDITED). If all pieces of P are M-crosscuts and cl q misses P u cl d_1 u cl d_2, then d_1, d_2 lie on one side of q. Sketch: else let kappa cross q once; win(Theta,.) differs by 1 on the two halves of q, yet equals a(kappa) near both ends (Z). SS1: a chain path with finitely many E-pieces lies on one side of every far crosscut. Used: D, R. D (LAB, UNAUDITED). No small E-crosscut has gamma and all gamma_d with finitely many E-pieces in total. Sketch: SS1 puts all pieces of gamma in Out(c) and all pieces of the second generation outside every E-piece of gamma; then each gamma_d contributes -eps(d) (E-piece) or 0 (M-piece, by CP1), so V2 = -V1 = +1. Used: (ii). K (LAB, UNAUDITED). For a chain path in a disk far from a point of X, the accumulation set of E-pieces is empty or perfect, with finitely many E-pieces in each gap. Sketch: an infinite run of consecutive E-pieces g_i would, by SS, lie in Sh(q_N) for a deep chain q_N -> x_0, so g_1 would lie in the intersection of the Sh(q_N), which is in X. Used: (iv). B (LAB, UNAUDITED). If pieces of P straddle a far crosscut q with ends x_0, x_1: P has a perfect set of E-pieces, Lambda_P(x_0) - Lambda_P(x_1) = +-1, and every continuum in X through the ends of P contains x_0 or x_1. Used: (c), (iv). AB (LAB, UNAUDITED; uses Newman, Elements of the Topology of Plane Sets V.14.3, cited from memory). P from u to v in a disk B, neighbouring crosscuts in B; if D_g(P) has one value for all deep E-crosscuts g near all points of X in 2B off P, then u, v are joined by a crosscut C in 2B with I(C,q) = I(P,q) for all E-crosscuts q far from 2B. Used: R (merging). D3 (LAB, UNAUDITED). If c, its E-pieces, and the merged crosscuts (an E-piece joined by AB with an adjacent M-only segment) all have first-generation arcs with finitely many E-pieces, then V2(c) = +1, a contradiction. So in (L2), for every eta > 0 some E-crosscut of diameter < eta has no such arc in B(h(a),eta/8); its arcs have perfect E-words straddling c. Used: context. R (LAB, UNAUDITED; check it). Let d_1..d_n be the E-pieces of gamma shallower than c (finite, sum eps = -1), P_0..P_n the segments (P_0 before d_1, P_n after d_n). Claim: under RIG, V2(c) = 1 + I(F o P_0, rho_c) + I(F o P_n, rho_c); under RIG + END*, V2(c) = +1, contradicting V2 in {0,-1}: no (L2) system exists. Lab proof: V2 = sum I(F o d_j, rho_c) + sum I(F o P_j, rho_c) (5.1). RIG at d_j with q = c makes the first sum -sum eps = +1. Interior P_j: RIG at c gives D_q(gamma) = -1 for far q deeper than c, RIG at the d_j gives D_q(P_j) = 0 for each j; AB joins the ends of P_j by an M-crosscut C_j; merging C_j with d_j gives d~, and RIG at d_j and d~ gives I(F o P_j, rho_c) = I(F o C_j, rho_c) = 0. (RIG is used at d_j, d~ (diam < 2r_1) and with q = c: scales must make them small and in Sh(c_*); INFERRED fixable by shrinking r_1 as in D3.) C. THE TARGET RIG: there is an E-crosscut c_* such that for every small E-crosscut c in Sh(c_*) and every E-crosscut q in Sh(c) with cl q n (cl c u F(cl c)) empty: a_c(q) = -1. (h once; no h^2, no finiteness.) END(c), lab form: for a first-generation arc gamma of c there are accessible points y_0 in P_0 n X near h(a) and y_n in P_n n X near h(b) such that D_q of the part of P_0 from y_0 to d_1, and of the part of P_n from d_n to y_n, is 0 for every far E-crosscut q (reading INFERRED; automatic if h(a), h(b) are accessible, or if gamma reaches them through M-pieces and arcs in X). What R needs (END*): I(F o P_0, rho_c) = I(F o P_n, rho_c) = 0. Known: the coherent case satisfies RIG; abstract word models (Cantor sets of re-entering pieces) violate it; for planar (L2) systems RIG is OPEN. Failure of RIG means: the image word of some E-crosscut has net E-data != -1 beyond the crosscut's own depth. D. ASK (a) Restate RIG, END, END* with all quantifiers (system, F, c_*, c, gamma, q). (b) DECIDE RIG for every (L2) system: PROVE it, REFUTE it by an explicit planar configuration consistent with B in which a_c(q) != -1 for arbitrarily deep q (name any (L2) hypothesis it fails), or REDUCE it to ONE smaller named lemma, all quantifiers, by a method not in F. (c) Prove END* automatic in (L2), or give the configuration where it fails (inaccessible h(a) approached by a perfect E-word). (d) Check R (the step "RIG at the d_j gives D_q(P_j) = 0", the AB merging, the scales); name the first false line. (e) Where planarity and nonseparation are used. (f) STILL OPEN EXACTLY: <= 3 statements, one next test each. E. CONTROLS (all must hold) (i) The coherent case (F o c homotopic to one E-crosscut shallower than c, run once) satisfies RIG. (ii) At a level of a planar periodic-point-free 2-adic tower (g cm, g^2 fixed-point free, not cm) V2 = +1 is allowed (INFERRED), so RIG must not by itself contradict anything there. (iii) X may contain no arc: use none. (iv) By K, a refutation needs a perfect E-word. (v) In R^3 crosscuts do not separate: which step fails? F. DEAD ENDS (not progress) Door-pocket image data; junction rules, product charts (vacuous on arcless hosts); chord-by-chord bookkeeping (self-similar profiles pass all laws); "each core window is one source depth interval" (FAILED); finite word enumerations as proof; "L2 iff two-generation finiteness at one c" (RESTATEMENT); local index, Cartwright-Littlewood (need h to extend). G. FORMAT Line 1 exactly: RIG-<PROVED|REFUTED-BY-configuration|REDUCED-TO-name|OPEN> ; END-<AUTOMATIC|FAILS-AT-configuration|OPEN> ; THMR-<HOLDS|BROKEN-AT-step|NOT-CHECKED>. Then SUMMARY (<= 100 words), the argument (<= 2,000 words), STILL OPEN EXACTLY. Short of effort: line 1 + summary; an honest OPEN is valid. No finite computation as proof; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists. RED FLAGS: h^2 cm used for RIG; a word model offered as planar; a scattered infinite word; data at q confused with data at c; arcs in X; h extended to the plane.