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Prompt

RESEARCH QUESTION (plane continuum theory). You are a research topologist. Label EVERY claim PROVED (you give the proof), SOURCE (exact reference and hypotheses; say if unchecked), DERIVED-BY-US (from the notes below, unaudited unless said), CONJECTURE or OPEN. Say plainly where your argument stops. A gap restated in new words is RESTATEMENT: say so. A short correct partial answer beats a long one. 1. SHARED SETTING PER = every homeomorphism of a nonseparating plane continuum has a periodic point. Known reduction (DERIVED-BY-US, modulo Blokh-Fokkink-Mayer-Oversteegen-Tymchatyn, Memoirs AMS 2013, "BFMOT"): L2 + "no planar 2-adic tower" implies PER, where L2 = "h^2 is never continuum-minimal for a periodic-point-free h". L2 is PROVED modulo BFMOT for a coherent outchannel; B1-B3 are three (not all) of the open residual cases. X is a nondegenerate nonseparating plane continuum (U = R^2 minus X connected). h : X -> X is a homeomorphism of X only, NOT assumed to extend to the plane. g is continuum-minimal (cm) if no proper nonempty subcontinuum K of X has g(K) = K (points count). Setting (S): h^2 is cm. Then h, h^2 are fixed-point free (m1 = min |h(x)-x| > 0, m2 likewise for h^2) and X is indecomposable, with empty interior, hereditarily unicoherent (DERIVED-BY-US, audited). F : R^2 -> R^2 is a perfect (continuous, proper) extension of h. Crosscut = open arc in U with ends in X; KP_delta = BFMOT KP-chords (circular crosscuts) of diameter < delta; var = BFMOT variation. By BFMOT (Thm 6.1.4, 6.2.1 as we read them) F has a unique outchannel E_t with principal continuum Pr(E_t) = X and variation -1. Its narrow strip is phi : [0,inf) x [-1,1] -> R^2, continuous, an embedding of the open strip into U: chords c_e = phi({e} x [-1,1]) with diam -> 0, sides p_e = phi(e,-1), q_e = phi(e,1), core kappa(e) = phi(e,0), depth e. The sides consist of chords in KP_delta and points in X (BFMOT Lemma 6.1.2). A small crosscut has var(F,.) = -1 if it crosses the external ray of E_t essentially, else 0. (XS): phi(e,+-1) lies in X for all e >= r1. (SC): the outchannel of an extension of h^{-1} is the same E_t (one dense channel). 2. BRANCH B1 = (NPC) [assume (S), (XS), (SC)] Product chart (PC): a closed disk B in R^2 with a homeomorphism Phi_B : B -> [0,1]^2, Phi_B(X cap B) = K_B x [0,1], K_B compact, 0,1 in K_B, K_B cap (0,1) nonempty (K_B = {0,1} is excluded), diam B < delta_c (a fixed small threshold). Mid band M_B = Phi_B^{-1}([0,1] x [1/4,3/4]). Convex chunk: K_B cap [x,y]. B1: for EVERY triple (X,h,F) with (S), (XS), (SC) there is NO PC. Known (DERIVED-BY-US, unaudited): W2: there are no PCs A, B with h(X cap M_A) = X cap M_B. Lemma P: let Fe(X) = points x with a neighbourhood in X homeomorphic to D x [0,1] (D Cantor), x off D x {0,1}; an (S) host has a PC iff Fe(X) is nonempty (via Tymchatyn-Walker 1998 Thm 2); Fe(X) is open and h-invariant. Theorem T (modulo W2): after straightening the image, h maps no small convex chunk of a PC onto a convex set. Box at x in Fe(X): an embedding beta : S x [0,1] -> X, S Cantor, beta(S x (0,1)) open in X, x = beta(c0,y0); strand c has ends b(c) = beta(c,0), t(c) = beta(c,1); for clopen I in S, Pi_I = partition of the ends of strands over I by the components of X minus beta(I x (0,1)). Oriented order near c0: (<,nu), < a linear order of a clopen neighbourhood of c0 inducing its topology, nu = +-1; circle word of a <-convex clopen I: b(c) (c increasing) then t(c) (c decreasing), nu = -1 swaps b,t; admissible: Pi_I non-crossing on the circle word of every <-convex clopen I. The planar strand order is admissible (disk lemma). CI (OPEN, h-free): X a nonseparating plane continuum, indecomposable, hereditarily unicoherent, empty interior; x in Fe(X); beta a box at x. Then ANY two admissible oriented orders near c0 have, inside every neighbourhood of c0, a nonempty clopen set convex for both. CI for all such (X,x) implies B1 (modulo W2). CI fails without indecomposability (Cantor fan). A counterexample to CI does NOT refute B1. 3. BRANCH B2 = SIDE CHORDS [neither (XS) nor (SC) assumed] Setting (L+): X as above, int X empty, g : X -> X a cm homeomorphism, g and g^2 fixed-point free (m1, m2), F, E_t, strip as above. Instance L2: g = h, h^2 cm. Instance TOWER (R2): a level (C_n, g_n) of a planar 2-adic tower (h periodic-point free; C_0 h-invariant; g_n = h^{2^n} on C_n is cm; C_{n+1} in C_n, g_n^2(C_{n+1}) = C_{n+1}, disjoint from g_n(C_{n+1})). (TSC) (NOT the (SC) of section 1) = a side channel: a positive-variation F^2-channel with g^2-invariant principal continuum P, door-hidden (its chain ends in one pocket) or one-wall (chain along one wall, P in limits of pocket continua). A DOOR is a chord s in KP_delta lying in a side, s = H_i((a_s,b_s)) (H_1(e) = p_e, H_2(e) = q_e), ends s-, s+ in X, depth window (a_s,b_s). Its POCKET Pk(s) is the component of U minus s missing the open strip; K_s = cl Pk(s) cap X (a continuum containing s-, s+: Janiszewski); pi(s) = diam cl Pk(s); s is MACROSCOPIC at scale eps if pi(s) >= eps. SCR(F,delta): for every r some door of the delta-strip has its window beyond r (= not (XS)). B2: for EVERY F and EVERY small delta, SCR(F,delta) fails (a) on EVERY L2-instance host; (b) on EVERY TOWER-instance host with (TSC) ((TSC) forces SCR, so (b) says (TSC) never occurs). Known (DERIVED-BY-US, unaudited): Lemma 0: if X contains no arc, SCR holds for every F and delta (no side is eventually constant, else Pr(E_t) is a point); so on arcless hosts B2 amounts to L2 resp. R2 for that host. Lemma D: pi(s) <= diam K + diam s for each continuum K in X joining s-, s+. Door laws: var(F,s) = 0; L2: var(F^2,s) in {0,-1}, -1 for at most one door; TOWER: a door hiding P has var(F^2,s) > 0 and pi(s) >= m2. (POCK-0) for a given (L+) system and F: "for every small delta, sup{pi(s) : window beyond r} -> 0 as r -> infinity". It would exclude the L2 instance (under (CD),(NG) of B3) and door-hidden (TSC); one-wall (TSC) would remain. (POCK-0) is OPEN. 4. BRANCH B3 = (DC-L2): outer-wall doors, L2 instance of B2 (g = h) Image arcs (audited lab block, modulo BFMOT): for deep r, F(c_r) is homotopic rel ends in a small disk about h(p_r) to a path gamma_r. (CD): gamma_r is a finite chain of whole chords c_{d_j}^{sig_j} (sig_j = +1: p to q; -1: q to p) joined by X-paths (junctions); a junction joins two points of one side iff its two pieces have opposite signs; also for door image paths (h(s-) to h(s+)). Data: extend kappa to (-inf,0] by an arc in U from infinity; for close x, y in X and far R, data(x,y)(R) = intersection number of a short path from x to y with the ray kappa((-inf,R]); data_D(R) = data(h p_D, h q_D)(R); for a door-free word, data_D(R) = sum of sig over the pieces of gamma_D of depth < R. Laws at every deep r: V1(c_r) = var(F,c_r) = sum_j sig_j [d_j < r] = -1; V2(c_r) = var(F^2,c_r) = sum_j sig_j data_{d_j}(r) in {0,-1}. A fold (a,b) = two pieces, consecutive once inner folds are removed, whose connecting X-path joins two points of a side (its junction side; the other side is its OUTER wall); their signs are opposite; its box is phi([d_a,d_b] x [-1,1]). (NG): every fold box of gamma_r and of the door paths met lies in B(h(p_r), rho_0), rho_0 < min(m1,m2)/4 fixed. Without doors a fold adds 0 to V1 and V2; the unpaired pieces all have sign -1, exactly one is shallower than r, and they add Ex >= 1 to V2. Theorem S (lab): (XS), (CD), (NG) contradict the laws. Door data D_s(R) = data(h s-, h s+)(R); it is 0 at far R unless pi(s) >= eps0 (eps0 = eps0(h) > 0). Pocket-winding rule: a door on the junction side of a fold is swallowed by the junction; for a fold (a,b) inside B(h(p_r), rho_0): data_{d_a}(r) - data_{d_b}(r) = sig_a sgn(d_b - d_a) sum of D_s(r) over doors s on its OUTER wall with window between d_a and d_b (sign: draft). B3: let (X,h,F) satisfy (S), (CD) and (NG) at all deep r. Then for every small delta there are NOT infinitely many pairwise distinct doors s_i of the delta-strip with pi(s_i) >= eps0, each on the outer wall of a fold of gamma_{r_i} (r_i -> infinity, both fold depths < r_i, window of s_i between them, D_{s_i}(r_i) != 0, door data cancelling Ex). (Theorem S' of the notes: an L2 counterexample with (CD),(NG) needs such doors.) Conjecture DP: the close points h^{-1}(s-), h^{-1}(s+) near p_r (joined by no small continuum: DERIVED-BY-US) are the ends of a macroscopic door within o(1) of c_r. Finite model (evidence only): only a macroscopic outer-wall door of a fold below the root survives (no door, small, junction-wall, too-deep doors: 0). Hand-check: r = 7, gamma_7 = (5,-)(2,-)(3,+) as (depth,sign), gamma_5 = (4,-), gamma_2 = (1,-), gamma_3 = (1,-)(6,-); fold (2,-)(3,+), junction side p; door s at depth 5/2 on its outer wall q, door path (6,-), so D_s(5/2) = 0, D_s(7) = -1; data_5(7) = data_2(7) = -1, data_3(7) = -2. Unaudited drafts: this example dies under a stricter location rule, larger ones survive; pulling a rescuing door back through an extension G of h^{-1} (F o G ~ id near X) forces a SECOND such door near c_r (door chains); ND ("for some eta0, e0 no door deeper than e0 has an eta0-dense pocket continuum") would suffice, but ND fails for general plane continua with a dense channel, so a proof must use h. 5. DEAD ENDS (do NOT repeat as progress) B1: scale-squeeze; descent over INFINITE chart sequences (also excludes h = id); dendrite fixed-point trap; Janiszewski or hereditary unicoherence inside a chart; countable invariant sets; order calculus alone (Cantor permutations are totally non-convex). B2, B3: hereditary-unicoherence junction rule, product charts (vacuous on arcless hosts); dendrite trap (needs (XS)); Tymchatyn-Walker Lemma 4; countable invariant sets (doors depend on F, delta); chord-by-chord bookkeeping alone (self-similar profiles pass all laws); (NG) from the pocket-winding rule. Not hosts: bucket handle, pseudo-arc (chainable); pseudo-circles (separate). 6. WHAT WE ASK Work on ONE branch of your choice (B3 = B2 (a) under (CD),(NG)). (a) Restate it with all quantifiers and the hypotheses of each lemma used. (b) PROVE, REFUTE, or REDUCE it to ONE smaller named lemma (all quantifiers) by a method NOT in section 5. (c) One control: B1: your argument must fail for h = id (PCs exist, only (S) fails); B2: Lemma 0; B3 the r = 7 hand-check (V1, V2, fold sign). (d) Where PLANARITY and NONSEPARATION are used; what fails in R^3. (e) STILL OPEN EXACTLY: <= 3 statements, each with one next test. A checkable counterexample is as welcome as a proof. 7. FORMAT (short, staged) Line 1 exactly: B1-NPC-<PROVED|REFUTED|REDUCED-TO-name|OPEN> ; B2-SIDE-<EXCLUDED|KEPT|REDUCED-TO-name|OPEN> ; B3-DC-L2-<EXCLUDED|KEPT|REDUCED-TO-name|OPEN> (Bk-NOT-ATTEMPTED if skipped). Then SUMMARY (<= 100 words), then the branch (total <= 2,000 words). If effort runs short, give line 1 and the summary: an honest OPEN is a valid answer. No finite computation as proof of an infinite statement; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists.