Transition Matrix without Continuation in the Conley Index Theory

The paper proves exactly the claim the candidate addresses: if a singular transition matrix entry T_n(p,q) is nonzero for an adjacent pair p∈P0, q∈P1, then there exists a sequence ε_m→0 such that, for each m, the extended slow–fast flow Φ^{ε_m}_t has a connecting orbit from M1_q×{1} to M0_p×{0}. The paper constructs the Morse decomposition of Ŝ_ε as M(S0)⊔M(S1) (Lemma 5.5.1), builds a connection matrix with block form whose off-diagonal T̂^ε encodes connections from λ=1 to λ=0, and defines the singular transition matrix via braid isomorphisms θ^0, θ^1: T_n(i,j)=θ^0_n(i)∘T̂_{n+1}(i,j)∘(θ^1_{n+1}(j))^{-1} (Definition 5.5.3). Nonzero T_n(p,q) forces nonzero T̂^{ε_m}_{n+1}(p,q) after passing to a subsequence with stable connection matrix, and Proposition 5.5.6 (together with the standard connection-matrix interpretation of the connecting homomorphism) yields the connecting orbit; this is formalized as Theorem 5.5.7 in the paper. These are the same ingredients and logic used by the candidate solution. Citations: the product-slice Morse decomposition (Lemma 5.5.1) is given in the paper’s Section 5.5, the block form of the connection matrix and degree conventions for T̂ appear just before the singular transition matrix definition, the definition T_n=θ^0∘T̂_{n+1}∘(θ^1)^{-1} is explicit, and the step “nonzero off-diagonal entry implies a connecting orbit” is the standard consequence of the connection matrix and connecting homomorphism (Proposition 4.2.7); the final implication is exactly Theorem 5.5.7. See , , , , and .