Reidemeister Traces of Iterates in Fixed Point Theory

• Reidemeister traces of iterates refine Lefschetz numbers by incorporating twisted conjugacy classes, offering a robust invariant for fixed point and periodic point analysis.
• The approach leverages bicategorical duality and zeta function formulations to decompose complex dynamical behaviors across algebraic topology and group theory.
• Its broad applicability in parametrized, equivariant, and quantum contexts provides actionable insights into homotopical and dynamical phenomena.

The Reidemeister trace of iterates is a central object in fixed point theory, algebraic topology, group theory, and dynamics, capturing subtle invariants associated with the periodic points of self-maps and their iterations. At the technical level, these traces refine the Lefschetz number by incorporating additional algebraic data—most notably, twisted conjugacy classes—and play a crucial role in the structure theory of fixed point invariants, zeta functions, and higher-categorical trace formalisms. Recent work has further illuminated their behavior in parametrized and equivariant settings, established deep connections with bicategorical duality, and revealed their role in quantum invariants and noncommutative geometry.

1. Algebraic and Topological Framework for Reidemeister Traces of Iterates

Given an endomorphism $f: X \to X$ of a topological space or a group endomorphism $\varphi: G \to G$, the $k$th iterate $f^k$ (or $\varphi^k$) leads to periodic phenomena naturally encoded by the sequence of Reidemeister traces $\{R(f^k)\}_{k\geq 1}$. For a discrete group $G$, the Reidemeister number $R(\varphi)$ is defined as the number of twisted conjugacy classes: $$x \sim_\varphi y \iff \exists\, z \in G: x = z y \varphi(z)^{-1}.$$ For maps on spaces, this algebraic construction models the topological partitioning of fixed or periodic point sets into homotopy-invariant classes, and $R(f^k)$ precisely counts essential fixed point (or periodic point) classes of $f^k$.

Analytically, the Reidemeister traces assemble into the Reidemeister zeta function,

$$R_\varphi(z) = \exp \left( \sum_{n=1}^{\infty} \frac{R(\varphi^n)}{n} z^n \right),$$

which encodes the periodic structure of the dynamics and links group-theoretic, homological, and dynamical information (Fel'shtyn et al., 2013, Fel'shtyn et al., 2014, Deré, 2023).

In the context of self-maps of manifolds, the Reidemeister trace $R(f^k)$ assigns to $f^k$ a global invariant living not in $\mathbb{Z}$ but in a more refined abelian group generated by twisted free loops (technically, the group ring $\mathbb{Z}[\pi_1 X_{f^k}]$, where $\pi_1 X_{f^k}$ is the fundamental group twisted by $f^k$) (Ponto et al., 2012).

2. Multiplicativity and Bicategorical Perspective

A key structural property of the Reidemeister trace is its multiplicativity under product and fibration structures, established via duality and traces in bicategories with shadow. For a fibration $p: E \to B$ with commuting endomorphisms $f: E \to E$ and $\overline{f}: B \to B$, the factorization theorem for iterates asserts

$$R(f^k) = [\widehat{R}_B(f)]^k \circ R(\overline{f}^k),$$

where $R(\overline{f}^k)$ encodes base fixed point class data (indices lifted to the group ring $\mathbb{Z}[\pi_1 B_{\overline{f}^k}]$) and $\widehat{R}_B(f)$ is the refined, fiberwise Reidemeister trace (Ponto et al., 2012). This bicategorical mechanism arises from the formal property

$$\operatorname{tr}(f \odot g) = \operatorname{tr}(g) \circ \operatorname{tr}(f)$$

for right-dualizable 1-cells $f$, $g$ in a bicategory with shadow, generalizing the classical symmetric monoidal trace. This ideology underpins not just multiplicativity but also additivity and decomposition of traces over colimits and diagrams (Ponto et al., 2014, Ponto et al., 2014).

Abstractly, iterated traces in 2-categories exhibit invariance properties: $1_X(1_Y(\varphi)) = 1_Y(1_X(\varphi))$ for composable dualizable 1-cells $X$, $Y$ and a 2-cell $\varphi$ (Campbell et al., 2019). This formalism supports Lefschetz theorems and Reidemeister trace computations for iterates and their higher-categorical generalizations.

3. Asymptotic, Dynamical, and Zeta Function Aspects

The sequence $\{R(\varphi^k)\}$ and the associated zeta function $R_\varphi(z)$ reflect both algebraic and dynamical complexity. For maps on infra-solvmanifolds and poly-Bieberbach groups, $R(\varphi^k)$ exhibits striking regularities:

• Under suitable hypotheses (e.g., all classes essential), $R(\varphi^k) = N(f^k)$, where $N(f^k)$ is the Nielsen number, and both sequences enter exponential zeta function formulas (Fel'shtyn et al., 2013, Fel'shtyn et al., 2014).
• The rationality of $R_\varphi(z)$ has been established for torsion-free virtually polycyclic (including virtually nilpotent and infra-nilmanifold) settings (Deré, 2023, Fel'shtyn et al., 2013); in general, it can fail if finiteness conditions are not met.
• The spectral radius $X(\varphi)$ (maximum modulus among eigenvalues contributing to $R(\varphi^k)$) determines the radius of convergence, and asymptotics of $R(\varphi^k)$ can be described as

$$R(\varphi^k) = \sum_{i=1}^{r(\varphi)} p_i \lambda_i^k$$

for algebraic numbers $\lambda_i$ and integer weights $p_i$, revealing periodic, quasi-periodic, or interval-like behavior among limit points (Fel'shtyn et al., 2014).

• Congruence relations exist: Gauss congruences of the form

$$\sum_{d|n} \mu(d) R(\varphi^{n/d}) \equiv 0 \pmod{n}$$

hold under finiteness and periodicity constraints, constraining the possible arithmetic and dynamical structure (Fel'shtyn et al., 2013).

Iterates also play a crucial role in periodic point theory. The height of an essential periodic orbit (the smallest $k$ for which it appears as an irreducible class) is governed by these traces, and analysis of their distribution informs the homotopy minimal period structure (Fel'shtyn et al., 2014).

4. Homotopical, Parametrized, and Equivariant Generalizations

The refinement of the Reidemeister trace to parametrized and equivariant contexts leverages the machinery of indexed monoidal categories and bicategories with shadow (Ponto et al., 2012). For fiber bundles $p: E \to B$ with fiberwise maps $f$, the fiberwise Reidemeister trace $R_B(f)$ and its iterates capture global fixed/periodic point information across the parameter space (Ponto et al., 2012).

A major development is the comparison with the fiberwise Fuller trace, which is a strictly more sensitive invariant than the collection $\{R_B(f^k)\}$ of fiberwise Reidemeister traces of iterates. Explicitly, for families $f: E \to E$ parametrized over $B$, it is possible for all fiberwise Reidemeister traces $R_B(f), R_B(f^2), \dots$ to vanish, while the parameterized Fuller trace $R_{B,C_n}(\Psi_B^n f)$ is nontrivial. This demonstrates the failure of the collection of iterated traces to detect all obstructions in parameterized settings, as proven by recent work resolving the conjecture of Malkiewich and Ponto (Williams, 25 Aug 2025).

Equivariant and bicategorical generalizations permit the analysis of iterated traces in spectral, equivariant, and stacky contexts, with additivity and locality results holding in parametrized stable homotopy theory (Ponto et al., 2014, Ponto et al., 2014). Such generalizations are essential for understanding the iterated trace invariants in higher-categorical and TQFT-inspired settings.

5. Reidemeister Traces of Iterates in Combinatorics, Torsion, and Quantum Topology

The theory extends beyond fixed point and periodic point counting: combinatorial formulations relate the traces of iterates to spanning tree enumeration and torsion invariants. For instance, the square of the Reidemeister torsion of a finite CW complex admits a spanning-tree expansion,

$$\tau^2(X; h) = \prod_{k \ge 0} \left( \sum_{T \in \mathcal{T}_{k+1}} w_T \right)^{(-1)^k}$$

where $w_T$ encodes combinatorial and homological data, and such formulas suggest connections between the fixed point data of iterates (e.g., on the mapping torus of $f$) and torsion computations (Catanzaro et al., 2012).

In quantum topology, the asymptotics of quantum invariants (e.g., colored Jones polynomials) for iterated torus knots reveal that leading coefficients are governed by twisted Reidemeister torsions associated to iterated structures, providing bridges between quantum invariants and classical fixed point data (Murakami, 2016).

Averaging formulas and decomposition theorems allow the computation of Reidemeister traces (and their iterates) via reductions to covering spaces, summing over coset representatives, and reducing higher-level invariants to manageable algebraic or combinatorial calculations—particularly effective for infra-nilmanifolds, torus bundles, and similar geometric situations (Dekimpe et al., 26 Sep 2024, Lee et al., 2016).

6. Key Formulas, Cyclicity, and Category-Theoretic Trace Calculus

Central formulas governing the theory include:

• For a fibration $p: E \to B$: $$R(f^k) = [\widehat{R}_B(f)]^k \circ R(\overline{f}^k)$$.
• Zeta function: $$R_\varphi(z) = \exp \left( \sum_{n=1}^{\infty} \frac{R(\varphi^n)}{n} z^n \right)$$.
• For diagrams in bicategories with shadow and pointwise dualizable objects:

$$\operatorname{tr}\big(\operatorname{colim}^\Phi(f)\big) = \sum_{[\alpha]} \phi_{[\alpha]} \operatorname{tr}(f_a \circ M_\alpha)$$

providing additivity for traces of iterates (Ponto et al., 2014, Ponto et al., 2014).

More categorically, the cyclic invariance of iterated traces, i.e.,

$1_X(1_Y(\varphi)) = 1_Y(1_X(\varphi)),$

forms the formal heart of many Lefschetz- and Reidemeister-type theorems, as traced in higher-categorical contexts (Campbell et al., 2019).

7. Open Problems and Future Directions

Several directions for further research are emphasized:

• Characterization of groups and endomorphisms for which $R(\varphi^k)$ is finite for all $k$, thus ensuring well-defined and rational zeta functions (Deré, 2023).
• Determination of conditions under which the additive and multiplicative decompositions described above extend to equivariant, parametrized, or non-orientable settings (Lee et al., 2016, Dekimpe et al., 26 Sep 2024).
• Exploration of the relationship between the periodic orbit structure (heights of irreducible classes), topological entropy, and spectral data for general endomorphisms (Fel'shtyn et al., 2014).
• Applications of iterated trace formalism in noncommutative geometry and stable homotopy theory, including the image of the cyclotomic trace and comparisons with topological Hochschild homology (Campbell et al., 2018).

This body of research establishes the Reidemeister trace of iterates as a deeply structured, categorically robust, and computationally powerful invariant, unifying themes from algebraic topology, categorical algebra, and dynamics, with ongoing impact in both geometry and higher algebra.