26 When counting errors imply mixing

26.1 A direct converse in the compact Selberg setting

There is a particularly close answer to the original trace-formula question. Let $X=\Gamma\backslash\mathbb H^2$ be a closed connected oriented surface of curvature $-1$. Let $\pi_X(L)$ count primitive hyperbolic conjugacy classes in $\Gamma$ of length at most $L$. Thus the two orientations are counted separately: $[\gamma]$ and $[\gamma^{-1}]$ are distinct. This is the primitive-class convention in the Selberg product; see [7, Section 2.1].

Two qualifications delimit this converse. First, (26.1) contains only the leading term: an error estimate after subtracting exceptional contributions $\operatorname{Li}(e^{s_jL})$ does not exclude those small eigenvalues. Second, the conclusion concerns the complete primitive spectrum. Applying it to simple curves or coefficient-filtered counts would require a corresponding spectral identity. If geometric geodesics are counted without orientation, the main term is $\tfrac12\operatorname{Li}(e^L)$; doubling that count recovers the stated convention. The exponent is quantitative, while explicit numerical or family-uniform mixing constants require additional estimates.

26.2 A quantitative converse in homogeneous dynamics

There is an established arithmetic converse. In the $S$-arithmetic setting of Gorodnik–Nevo [10, Theorems 1.4 and 5.2], let $E_{S,m}(r,B)$ denote the $L^2$ discrepancy of normalized rational point counts in translated real balls of radius $r$ and an $S$-adic set $B$. For a discretely embedded irreducible representation $\pi$ in the mean-zero automorphic spectrum, their estimate is $$\|\pi(\beta_B)\| \leq 2[\Gamma_S:\Gamma_{S,m}] m_{G_\infty}(B(e,r))^{-1}E_{S,m}(r,B), \qquad 0<r\leq r_0(\pi).$$(26.5) The ambient group and congruence quotients satisfy the standing algebraic hypotheses of that paper, including a connected, simply connected $\mathbb Q$-simple group in the notation displayed here. Here the real factor is noncompact, the places in $S$ are unramified, $B$ is bounded, positive-measure, and bi-invariant under the specified hyperspecial compact subgroup, and $\beta_B$ is its normalized Haar measure. The small-radius threshold depends on $\pi$. Thus sufficiently strong localized discrepancy controls averaging operators quantitatively. Their application to forms of $\mathrm{SL}_2$ uses arithmetic counting to obtain a spectral gap. This statement is not a general implication from one scalar count to mixing of a flow.

26.3 A converse for a finite Markov model

Let $A$ be an irreducible nonnegative integer $q\times q$ adjacency matrix, allowing parallel edges, and let $\lambda>0$ be its Perron eigenvalue. The number $F_n$ of periodic edge sequences fixed by the $n$th shift, with a distinguished starting position, is $$F_n=\mathop{\mathrm{tr}}(A^n).$$ It includes repetitions and is not the number of primitive orbits. The associated identity $$\zeta_A(z)= \exp\left(\sum_{n\geq1}\frac{\mathop{\mathrm{tr}}(A^n)}n z^n\right) =\frac1{\det(I-zA)}$$ is the usual finite-state trace formula; see [11, Sections 1 and 4].

To convert the gap into a mixing statement, choose positive Perron vectors $r,l$ normalized by $\sum_i l_ir_i=1$, and set $$P_{ij}=\frac{A_{ij}r_j}{\lambda r_i}, \qquad \pi_i=l_ir_i,\qquad \Pi=\mathbf1\pi^T.$$ Then $P$ is stochastic, $\pi$ is stationary, and $P$ is similar to $A/\lambda$. For every $0<\kappa'<\kappa$, $$\|P^n-\Pi\|_\infty\leq C_{\kappa',A}e^{-\kappa'n}.$$(26.8) At the endpoint $\kappa$ a polynomial factor can arise from Jordan blocks. In particular, the stationary Markov chain satisfies $$\left|\mathbb E_\pi[f(X_0)g(X_n)] -\mathbb E_\pi f\,\mathbb E_\pi g\right| \leq C_{\kappa',A}e^{-\kappa'n}\|f\|_\infty\|g\|_\infty.$$ Fixed-width cylinder observables have the same decay after the corresponding time offset. Approximation by cylinder functions gives mixing of the shift, and quantitative approximation gives a rate for Hölder observables with an explicit regularity loss. For example, on the two-sided stationary vertex shift use the metric $d_\theta(x,y) =\theta^{N(x,y)}$, where $0<\theta<1$ and $N(x,y)$ is the least $|j|$ for which $x_j\ne y_j$. A $\beta$-Hölder test is approximated by conditioning on coordinates $-p,\ldots,p$ with error $O(\theta^{\beta p})$. The Markov property bounds the covariance of the two resulting cylinder tests at lag $n>2p$ by $O(e^{-\kappa'(n-2p)})$ times their sup norms. Taking $p=\lfloor\kappa'n/(2\kappa'+\beta|\log\theta|)\rfloor$ gives the rate $$\exp\left(-\frac{\kappa'\beta|\log\theta|} {2\kappa'+\beta|\log\theta|}\,n\right).$$ Parallel-edge observations are handled by conditioning on the adjoining vertices; a bounded change in the time offset changes only the constant.

The matrix constant can itself be bounded explicitly. If $r_0=e^{-\kappa}<r<1$, the resolvent integral around $|z|=r$ gives $$\|P^n-\Pi\|_\infty \leq \frac{r\,q!(1+r)^{q-1}} {(1-r)(r-r_0)^{q-1}}\,r^n.$$(26.9) Indeed, on this circle the determinant has modulus at least $(1-r)(r-r_0)^{q-1}$, while the row-sum norm of its adjugate is at most $q!(1+r)^{q-1}$. Thus a numerical counting saving produces a numerical mixing estimate in this model.

26.4 Which scalar count is being estimated?

An exact count of unrestricted paths need not imply mixing. For example, with $m\geq2$, $$A=\begin{pmatrix}0&m\\m&0\end{pmatrix}, \qquad \mathbf1^T A^n\mathbf1=2m^n.$$ This count has no error at all. Its normalized transition chain alternates between two states; the correlation of $(1,-1)$ with itself is $(-1)^n$. The closed-path count detects what the total path count misses: $$\mathop{\mathrm{tr}}(A^n)=m^n+(-m)^n.$$ Thus a scalar sequence can suffice when it is a complete trace encoding of a suitable operator. The decisive issue is what the chosen count retains.

Continuous time introduces a separate return-time issue. The full shift on $m$ symbols has exactly $m^n$ fixed points for its $n$th iterate and is mixing for its Bernoulli measure. Suspend it with constant roof $1$: $$Y=(\Sigma\times[0,1])/ \bigl((x,1)\sim(\sigma x,0)\bigr).$$ For the suspension flow $\phi^t$, the mean-zero function $F([x,u])=e^{2\pi i u}$ is well defined and satisfies $$F\circ\phi^t=e^{2\pi it}F,\qquad \int\overline F\,F\circ\phi^t\,\mathop{}\!\mathrm d\nu=e^{2\pi it}.$$ The flow is therefore not mixing. Its physical periods lie in a lattice, so this is not an example of an unoscillating continuous-time prime-orbit asymptotic. It shows why a discrete-time counting converse does not automatically pass to a suspension. Ruling out arithmetic return times is necessary; quantitative flow estimates also require control of approximate eigenfunctions or high-frequency return-time operators. The role of these additional geometric conditions is discussed in [13].

26.5 A qualitative converse from full geodesic counting

There is also a simple geometric deduction. In Babillot [12, Theorem 1], the universal cover is a simply connected complete Riemannian manifold with sectional curvature bounded above by a negative constant, and the acting discrete isometry group is non-elementary. A quasi-product measure means a flow-invariant measure whose geodesic current is in the measure class of a product of its two boundary measures. For such a measure, nonarithmeticity of the closed length spectrum implies mixing. Assume a nonzero finite such measure, normalized to probability, and suppose the full primitive closed-orbit count is finite at each cutoff and satisfies $$\pi(T)\sim C\,\frac{e^{hT}}{T},\qquad C,h>0,$$ as $T\longrightarrow\infty$ through all real values. If all periods belonged to $a\mathbb Z$, $a>0$, then $$\pi(na+a/4)=\pi(na+3a/4),$$ whereas the ratio of their proposed main terms tends to $e^{ha/2}>1$. This contradiction proves nonarithmeticity and hence qualitative mixing under the stated structural hypotheses. No rate follows from this argument. For a polynomial simple-curve main term $CT^d$, the corresponding fixed-shift ratio tends to one, so this argument does not transfer to simple-curve counting.

26.6 Localized counts and correlations

A useful general converse applies once geometric counts are related to joint transition masses. Let $(Y,d,\mu)$ be a compact metric probability space, with a measure-preserving flow $\phi^t$. Suppose that for every $0<h\leq1$ there is a measurable partition $\mathcal P_h=\{P_1,\ldots,P_{N(h)}\}$ with $$\operatorname{diam}(P_i)\leq h,\qquad N(h)\leq B h^{-D}.$$ Zero-measure cells may be discarded. Define $p_{ij}(t,h)=\mu(P_i\cap\phi^{-t}P_j)$.

If the right side of (26.10) contains the additional factor $\mu(P_i)\mu(P_j)$, its sum costs one, and the exponent improves to $\kappa\beta/(m+\beta)$. The normalization of a localized count therefore materially affects the resulting rate.

The substantive hypothesis is the geometric identification of counts with $p_{ij}$. In an application one needs a formula of the form $$p_{ij}(t,h)= \frac{\text{weighted segment count from }P_i\text{ to }P_j} {\text{normalizing growth}} +\text{controlled error}.$$ Jacobian weights, multiplicities, boundary effects, and time-window errors must be controlled uniformly at both endpoints and at shrinking resolution $h$. If the comparison error satisfies the same bound as the normalized counting error, their constants add and the proposition applies. The proposition does not supply this comparison for an arbitrary raw counting problem.

26.7 Nonescape and the counting time parameter

There is an explicit version with a noncompact tail. Suppose an exhaustion $K_H$ satisfies $\mu(Y\setminus K_H)\leq C e^{-aH}$, with $a>0$. Assume that $K_H$ has at most $B e^{bH}h^{-D}$ cells of diameter $h$, and that the entrywise bound is $A e^{cH}h^{-m}e^{-\kappa t}$, where $b,c,m\geq0$. For bounded globally $\beta$-Hölder observables the preceding argument, extending the step approximations by zero off $K_H$, gives $$|\operatorname{Corr}_t(f,g)| \leq C\|f\|_{C^\beta}\|g\|_{C^\beta} \left(h^\beta+e^{-aH} +e^{(2b+c)H}h^{-(m+2D)}e^{-\kappa t}\right).$$ The tail error uses invariance and the $L^1$ approximation bound; no uniform approximation on the discarded region is required. Taking $h=e^{-\eta t/\beta}$ and $H=\eta t/a$ yields $$\boxed{\displaystyle \eta= \frac{\kappa} {1+(2b+c)/a+(m+2D)/\beta}.}$$(26.12) These are declared sufficient hypotheses, not estimates established here for Teichmüller flow.

Finally, under the counting-time conversion $R=e^t$, a relative power saving becomes an exponential one: $$\frac{O(R^{d-\delta})}{R^d}=O(e^{-\delta t}).$$ This explains why the exponents in effective counting and mixing are related. The conversion alone gives no transition-mass identity. A converse for our curve-counting program would require the estimates to retain independent endpoint or sector weights, with controlled dependence on localization and on excursions into the noncompact part of the parameter space. Alternatively, an appropriate trace or transfer-operator representation could provide a spectral converse. Neither representation follows merely from a good remainder for the single unweighted simple-length count.