Time-Dependent Birkhoff Sums in Dynamics

• Time-dependent Birkhoff sums are partial ergodic sums defined with a time parameter, revealing fluctuations beyond classical spatial averages.
• They employ renormalization and multifractal analysis to capture self-similarity, deviation phenomena, and singular behavior in low-dimensional rotations.
• The study highlights nonstandard limit laws, including the breakdown of CLT, anomalous scaling, and extreme oscillatory features in various measure regimes.

Time-dependent Birkhoff sums are partial ergodic sums in deterministic or random dynamical systems, evaluated as functions of a "time" parameter rather than as uniform Cesàro averages. Unlike classical Birkhoff sums, whose asymptotics are driven by space averages and averaged ergodic theorems, time-dependent or "quenched" Birkhoff sums probe the fluctuating, non-averaged regimes—revealing multifractal statistics, self-similarity, deviation phenomena, and nonstandard limit laws as the summation endpoint evolves. This article synthesizes recent advances in the rigorous study of time-dependent Birkhoff sums, with special focus on rotations and low-dimensional systems, non-integrable observables, renormalization structures, and oscillatory regimes outside the classical ergodic paradigm.

1. Definition and General Framework

Let $(X, \mathcal{F}, \mu)$ be a probability space and $T: X \to X$ a measure-preserving transformation. For an observable $f: X \to \mathbb{R}$, the $n$-th Birkhoff sum is

$S_n(f, x) = \sum_{j=0}^{n-1} f(T^j x).$

In the time-dependent context, one regards $S_{n(t)}(f, x)$ as a function of a generic time parameter $t$, with $n(t)$ increasing (often taken to be $[t\cdot N]$ or along explicit integer subsequences). Variants include the case where the transformation $T$ or the observable $f$ evolves with $n$, producing non-autonomous or random compositions. The focus is on the fine statistics and asymptotic laws of these sums as $t \to \infty$, possibly along sparse or renormalization-tuned subsequences.

For the canonical case of irrational rotations $T_\alpha(x)=x+\alpha \pmod{1}$ on $\mathbb{T}^1$, time-dependent sums are

$$S_N(f, \alpha, x_0) = \sum_{n=1}^N f(n\alpha + x_0).$$

These serve as a testbed for the subtle phenomena that arise when the observable $f$ is singular (e.g., unbounded variation, logarithmic or Cauchy-type singularities) or for irregular arithmetic parameters $\alpha$.

2. Self-Similarity and Renormalization for Singular Observables

A central case involves the Birkhoff sums of observables with logarithmic or power-law singularities evaluated at diophantine rotations. Knill and Tangerman (Knill et al., 2010) analyze

$$S(k,\alpha) = \sum_{j=1}^k g(j\alpha) \quad \text{with} \quad g(x) = \log(2-2\cos(2\pi x))$$

for the golden-mean rotation $\alpha = (\sqrt{5}-1)/2$, which arises in critical KAM theory. Key structural findings include:

• Asymptotic Growth: Despite $g \notin BV$, truncation methods yield $|S(k,\alpha)| \leq C (\log k)^2$ and, more sharply,

$$0 \leq \liminf_{k\to \infty} \frac{S(k,\alpha)}{\log k} \leq \limsup_{k\to\infty} \frac{S(k,\alpha)}{\log k} = 2,$$

and numerical experiments show the distribution of $\frac{S(k,\alpha)}{\log k}$ is supported on $[0,2]$.

• Continued-Fraction and Return Times: At times $k=q_n$ (Fibonacci sequence), the sum stabilizes in a renormalized sense:

$$S(q_n-1,\alpha) = 2\log q_n + O(1),\qquad S(q_n,\alpha) \to L < \infty.$$

• Renormalization Limit Function: For $x \in [0,1]$,

$$f_n(x) = S(\lfloor x q_n\rfloor, a_{n+1}) - S(\lfloor x q_n\rfloor, a_n)$$

converges almost everywhere to a limit $f(x)$ solving the self-similarity (cohomological) equation

$f(\alpha x) + \alpha^2 f(x) = \beta(x),$

where $\beta$ is a staircase-type forcing function.

• Macroscopic and Microscopic Scales: The fine structure of $S(k,\alpha)$ decomposes as

$$S(k, \alpha) = h\left(\frac{k}{q_n}\right) + S\left( \left\lfloor \frac{k}{q_n} \right\rfloor, a_n \right), \quad \text{with}\quad h(x)=\sum_{j=0}^\infty f(\alpha^j x),$$

and the ultimate limit

$L = \log\left(\frac{4\pi^2}{5}\right) + h(1^-).$

Self-similarity, encoded in the functional equation for $f$, accounts for the observed nontrivial limiting distribution and slow oscillations. Related self-similar attractors appear for cotangent observables (Knill, 2012).

3. Distributional Limits, Oscillations and Absence of Standard CLT

Time-dependent Birkhoff sums of bounded variation observables for circle rotations display a mixture of regular (concentrated) and irregular (oscillatory, anti-concentrated) behavior depending on arithmetic and functional context:

• Central Limit Theorem (CLT) Regimes: For quadratic irrationals or badly approximable $\alpha$ and $f$ of bounded variation (e.g., indicator or sawtooth functions), temporal CLTs with variance scaling $O(\log N)$ hold for the normalized family $\left\{ S_N(f,\alpha) : N \leq M\right\}$, yielding Gaussian limiting laws under temporal sampling (Borda, 2021, Borda, 2023). See Table:

Regime	Scaling of $\mathrm{Var}$	Limiting Law
Quadratic/BAdly Approx	$\log N$	Gaussian/Stable
Typical $\alpha$	$\log N\,\log\log N$	No CLT, heavy tails

• No-CLT Phenomena: For almost every $\alpha$, there does not exist a temporal distributional limit theorem for piecewise smooth zero-mean $f$, no matter how the centering and scaling sequences are chosen (Frühwirth et al., 2023). This rigidity is caused by persistent oscillations introduced by discontinuities or singularities—two distinct subsequential limiting laws emerge, precluding convergence along the full sequence. As soon as $f$ has a discontinuity or singularity, and for almost every $\alpha$, the normalized Birkhoff sums do not converge in distribution.
• Extreme Value Laws: Recent work establishes the limit law (1-stable) for the maximum of the Birkhoff sum for generic observable and random $\alpha$, with the maximal deviation scaling as $\log N \log\log N$ (Borda, 2023). This reflects the absence of concentration typical for i.i.d. sums, illustrating the unique effect of deterministic dynamical correlations.

4. Growth, Oscillation, and Anomalous Scaling in Infinite-Measure and Nonintegrable Cases

In infinite-measure dynamical systems or when the observable $f$ is nonintegrable, Birkhoff sums display anomalous scaling and oscillatory growth profiles:

• Infinite-Mean Case: For observables $\varphi\geq 0$ with heavy tails $\mu(\varphi \geq u) \sim u^{-\alpha}$ ($0<\alpha<1$),

$$S_n \varphi(x) \sim n^{1/\alpha} \quad \text{(almost surely upper/lower bounds),}$$

as shown in (Galatolo et al., 2018) under super-polynomial decay of correlations or Gibbs–Markov structure.

• Oscillation in Weak Mixing: Without sufficient decorrelation, wild oscillation between different scaling rates can occur, resulting in $\limsup \neq \liminf$ for logarithmic growth exponents of $S_n\varphi(x)$.
• Extreme Value Analogues: The behaviors of partial maxima and record statistics for time-dependent Birkhoff sums are tightly connected to hitting times and run-length phenomena, with strong deviations from classical Borel–Cantelli (see Section 4 of (Galatolo et al., 2018)).

5. Central Limit Theorems for Time-Dependent Dynamical Systems

In non-autonomous, random, or quasistatic dynamical settings, the asymptotic behavior of time-dependent Birkhoff sums is governed by a combination of spectral properties, variance growth, and mixing rates:

• Variance Growth and Coboundary Conditions: The central limit theorem (CLT) for Birkhoff-like sums requires that the variance diverges. For time-dependent compositions $T_n$ on $(X,\mathcal{F},p)$ and centered observable $f_n$,

$$S_N(x) = \sum_{n=0}^{N-1} f_n\left(T_{n-1} \circ \cdots \circ T_0(x)\right),$$

the sufficient condition is the "accumulated transversality" or absence of a time-dependent cohomological solution (Nandori et al., 2011).

• Rates and Correlations: The optimal Berry–Esseen rate $O(N^{-1/2})$ or $O(N^{-1/2}\log N)$ can be proven, with explicit constants, using functional correlation decay and Stein’s method (via Sunklodas' approach), provided scaling sequences for covariance growth and mixing are controlled (Leppänen et al., 2019, Hella et al., 2018).
• Intermittent Maps: Time-dependent compositions of Pomeau–Manneville maps again yield self-normalized CLTs, with polynomial rate depending on the maximal intermittency exponent (Hella et al., 2018). The variance growth exponent controls the regime where the CLT applies.

6. Fluctuations, Fast/Slow Points, and Multifractal Growth

The time-dependent framework supports a multifractal analysis for growth rates and limsup assumptions:

• Fast and Slow Point Dichotomy: For a uniquely ergodic $T$ and $f$ of zero mean, the growth rate of $|S_n f(x)|$ varies between points: for a residual set of $f$ (Baire generic), almost every $x$ exhibits arbitrarily fast sublinear growth, while almost every $(\alpha, x)$ in measure exhibits at most $O(n^{1/2+\varepsilon})$ scaling (Bayart et al., 2019).
• Hausdorff Dimension: The exceptional "slow point" sets (where $|S_n f(x)|=O(\psi(n))$) for $\psi(n)=o(n)$ are shown to be of zero Hausdorff dimension for generic $f$ (Bayart et al., 2019).
• Arithmetic and Discrepancy Connections: The entire time-dependent fluctuation regime is tightly linked to the fine structure of the rotation number $\alpha$ and its continued-fraction expansion, with deep links to discrepancy theory (see, e.g., (Ralston et al., 27 Nov 2025) for an explicit connection between the support of Birkhoff measure and discrepancy).

7. Broader Implications and Open Directions

Time-dependent Birkhoff sums reveal that dynamical systems that are highly regular in ergodic mean can admit complex, oscillatory, and often fractal time-profile behavior when examined pointwise in time. Key consequences and open problems include:

• Renormalization and Multiscale Attractors: Self-similarity equations and renormalization group techniques provide a precise universal structure for singular observables at diophantine rotations, linking to number-theoretic symbolic codings (e.g., $\beta$-expansions in (Knill, 2012)).
• Quantum Modular Phenomena: Birkhoff sums also provide explicit bridges to "quantum modular forms," whose discontinuous yet a.e. continuous cocycles parallel the discontinuous limits of time-dependent ergodic sums (Borda, 2023).
• Limit Distribution Classification: The existence or breakdown of distributional limit theorems is finely tuned by the observable's regularity and the arithmetic class of the system. The universality of "no-TDLT" (temporal distributional limit theorem) regimes for discontinuous observables is now established (Frühwirth et al., 2023).
• Infinite-Measure and Heavy-Tailed Dynamics: Anomalous scaling, excessive oscillations, and connections to recurrence and local times are now quantitatively characterized for a broad class of infinite-measure and nonintegrable observables (Galatolo et al., 2018, Pène, 2023).
• Outstanding Questions: The extension to higher-dimensional actions, the full classification of observables for which temporal limit laws exist, and the multifractal/statistical interpretation of time-dependent attractors remain active topics across ergodic theory, number theory, and mathematical physics.

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References:

• "Selfsimilarity and growth in Birkhoff sums for the golden rotation" (Knill et al., 2010)
• "On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational" (Frühwirth et al., 2023)
• "Limit laws of maximal Birkhoff sums for circle rotations via quantum modular forms" (Borda, 2023)
• "Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables" (Galatolo et al., 2018)
• "Fast and slow points of Birkhoff sums" (Bayart et al., 2019)
• "Selfsimilarity in the Birkhoff sum of the cotangent function" (Knill, 2012)
• "A central limit theorem for time-dependent dynamical systems" (Nandori et al., 2011)
• "Central limit theorems with a rate of convergence for time-dependent intermittent maps" (Hella et al., 2018)
• "Sunklodas' approach to normal approximation for time-dependent dynamical systems" (Leppänen et al., 2019)
• "Birkhoff Measures, Birkhoff Sums, and Discrepancies" (Ralston et al., 27 Nov 2025)
• "On the distribution of Sudler products and Birkhoff sums for the irrational rotation" (Borda, 2021)
• "Limit theorems for Birkhoff sums and local times of the periodic Lorentz gas with infinite horizon" (Pène, 2023)