Collatz conjecture and symbolic dynamics
I study the Collatz conjecture by combining number theory and symbolic dynamics. The central idea is to encode the orbits of the accelerated map $T$ as words over a finite alphabet and as paths in a directed graph modulo 6: each integer becomes a sequence of symbols and each iteration an admissible transition of the graph. An arithmetic question thus turns into the study of a symbolic system —a subshift of finite type—, where the transition graph, its strongly connected components and the admissible words yield exact results: the Fibonacci theorem for trajectories, the fiber-growth dichotomy, and reductions of the conjecture in terms of the residue $h_m(n) \in \{0,\ldots,5\}$. The long-term goal is to fully characterize the cycles of $T$ and of its generalizations $T_D$ through this symbolic lens.
Paper 1
A Fibonacci theorem for Collatz trajectories via modular graph structure
Proves that for each $m\ge1$, exactly $F(m+1)$ odd integers in $\{1,\ldots,2^m\}$ have orbits under $T$ that avoid residue class $4\pmod{6}$ during steps $2,\ldots,m$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number. The proportion decays at rate $(\varphi/2)^m$. The proof uses the directed graph $G$ of Collatz transitions mod 6 and its unique absorbing strongly connected component. An explicit bijection is constructed encoding integers as directed paths in $G$, and it is shown that every positive cycle of $T$ must visit residue class $2\pmod{6}$, which by a flow conservation identity accounts for more than 18 % of the steps.
Research lines
Symbolic–modular encoding
Bijections $\Phi_m$ (binary) and $\Psi_m$ (mod 6), graph $G$ mod 6 and its strongly connected components. Structural backbone of all three papers.
Fiber growth and strong couplings
Fiber-growth dichotomy controlled by $h_m(n)$, canonical injection between translated fibers, and reduction of the conjecture to Hamming weight.
Conjecture $h_m = 4$ and descent
Residue 4 mod 6 as the modular signature of every strong coupling. Descent lemma and its implications for the uniqueness of cycles.
Generalized Collatz and cycles
Family $T_D$ for odd $D$, universal rotation–cycle correspondence, and Moreau's necklace formula for counting length-$m$ cycles of weight $s$.
Visualizations
Bijections Φm and Ψm
Explore the binary and mod 6 encoding of Collatz orbits: periodicity, antisymmetry and the closed-form formula for Tm(n).
View visualizations →Publications
A Fibonacci theorem for Collatz trajectories via modular graph structure
Manuel-Alejandro Reyes Jiménez — math.NT — 28 May 2026
Conferences
Participation in conferences and seminars will be posted here as it happens.