Arithmetic Hyperbolic Dynamics Visualiser

The hyperbolic plane: upper half-plane and disk

The upper-half-plane model is \[ \mathbb H=\{z=x+iy\in\mathbb C:y>0\}, \qquad ds_{\mathbb H}^2=\frac{dx^2+dy^2}{y^2}, \qquad dA_{\mathbb H}=\frac{dx\,dy}{y^2}. \] Its geodesics are vertical Euclidean lines and Euclidean semicircles perpendicular to the real axis.

The Poincaré disk is \[ \mathbb D=\{w\in\mathbb C:|w|<1\}, \qquad ds_{\mathbb D}^2=\frac{4|dw|^2}{(1-|w|^2)^2}. \] Its geodesics are diameters and circular arcs meeting the unit circle orthogonally.

The two pictures are exactly isometric

The Cayley transform \[ C:\mathbb H\longrightarrow\mathbb D, \qquad C(z)=\frac{z-i}{z+i}, \] has inverse \[ C^{-1}(w)=i\,\frac{1+w}{1-w}. \] It sends \(i\mapsto0\), the boundary \(\mathbb R\cup\{\infty\}\) to the unit circle, and pulls the disk metric back to the upper-half-plane metric: \[ C^*(ds_{\mathbb D}^2)=ds_{\mathbb H}^2. \] Therefore the half-plane and disk displays in the app are two coordinate pictures of the same hyperbolic point, geodesic, horocycle, Hecke correspondence, and quotient geometry.

The Cayley transform also carries horocycles to horocycles: Euclidean circles tangent to the boundary (with horizontal lines viewed as circles tangent at \(\infty\)) become Euclidean circles tangent to \(\partial\mathbb D\). Thus changing model never changes the underlying dynamics.

A useful distance formula in the upper half-plane is \[ \cosh d_{\mathbb H}(z,w) = 1+\frac{|z-w|^2}{2\,\operatorname{Im}z\,\operatorname{Im}w}. \] The app uses hyperbolic rather than Euclidean geometry when it compares quotient representatives.

\(\mathrm{PSL}_2(\mathbb R)\) as the orientation-preserving isometry group

For \[ g= \begin{pmatrix}a&b\\c&d\end{pmatrix} \in\mathrm{SL}_2(\mathbb R), \] the Möbius action \[ g\cdot z=\frac{az+b}{cz+d} \] preserves \(\mathbb H\) and its hyperbolic metric. The matrices \(g\) and \(-g\) induce the same Möbius transformation, so \[ \operatorname{Isom}^+(\mathbb H) \cong \mathrm{PSL}_2(\mathbb R) = \mathrm{SL}_2(\mathbb R)/\{\pm I\}. \]

The stabiliser of \(i\) is the compact subgroup \[ K=\mathrm{PSO}(2), \] and hence \[ \mathbb H \cong \mathrm{PSL}_2(\mathbb R)/K. \] A point of \(\mathbb H\) remembers position only. A point of the unit tangent bundle remembers position and direction, giving \[ T^1\mathbb H\cong\mathrm{PSL}_2(\mathbb R). \]

If \(\Gamma<\mathrm{PSL}_2(\mathbb R)\) is discrete, then \[ X=\Gamma\backslash\mathbb H \] is a hyperbolic surface or orbifold, while \[ T^1X \cong \Gamma\backslash\mathrm{PSL}_2(\mathbb R). \] AQUE is naturally formulated on this unit tangent bundle because semiclassical limits retain both position and momentum direction.

Right \(K\)-motion versus left \(K\)-motion

Right multiplication \(g\mapsto gk\) leaves the base point \(g\cdot i\) fixed and rotates the tangent vector. By contrast, left multiplication \(g\mapsto kg\) generally moves the base point; geometrically it rotates \(\mathbb H\) about \(i\).

One-parameter subgroups, flows, and standard decompositions

With \[ a_t= \begin{pmatrix}e^{t/2}&0\\0&e^{-t/2}\end{pmatrix}, \qquad u^+(s)= \begin{pmatrix}1&s\\0&1\end{pmatrix}, \qquad u^-(s)= \begin{pmatrix}1&0\\s&1\end{pmatrix}, \] right multiplication by \(a_t\) is geodesic flow, right multiplication by \(u^+(s)\) is the stable horocycle flow in the convention of this app, and right multiplication by \(u^-(s)\) is the unstable horocycle flow.

Stability follows from conjugation: \[ a_{-t}u^+(s)a_t=u^+(e^{-t}s), \qquad a_{-t}u^-(s)a_t=u^-(e^{t}s). \] Hence a small \(U^+\)-displacement contracts in forward geodesic time, whereas a small \(U^-\)-displacement expands.

The compact subgroup \(K=\mathrm{SO}(2)\) supplies the fourth real motion in the Dynamics menu. Its right action rotates the tangent vector without moving the base point. The app's displayed tangent-angle convention uses \(g\mapsto gk_{t/2}\) when the displayed angle changes by \(t\).

Iwasawa decomposition — geometric coordinates

Every \(g\in\mathrm{PSL}_2(\mathbb R)\) has an Iwasawa decomposition \[ g=n_xa_yk_\theta, \qquad n_x=\begin{pmatrix}1&x\\0&1\end{pmatrix}, \quad a_y=\begin{pmatrix}\sqrt y&0\\0&1/\sqrt y\end{pmatrix}. \] Since \(k_\theta\cdot i=i\), \[ g\cdot i=x+iy. \] Thus \((x,y,\theta)\) are position-plus-direction coordinates on \(T^1\mathbb H\).

Cartan decomposition — hyperbolic polar coordinates

Every \(g\) can also be written \[ g=k_1a_tk_2,\qquad t\ge0. \] With the normalisation above, \[ t=d_{\mathbb H}(i,g\cdot i). \] Thus \(KAK\) is the group-theoretic form of geodesic polar coordinates; algebraically it is closely related to singular-value decomposition.

Bruhat decomposition — two algebraic cells

If \(B=AU^+\) is the upper-triangular Borel subgroup and \[ w=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \] then \[ \mathrm{PSL}_2(\mathbb R)=B\sqcup BwB. \] Bruhat decomposition is fundamental in automorphic forms and double-coset calculations, although it is less directly responsible for the entropy mechanism than the \(U^-AU^+\) local product structure.

Finally, the local product \[ U^-AU^+ \] is the decomposition most directly relevant to stable/unstable dynamics, Bowen balls, entropy, and the Bourgain–Lindenstrauss small-box estimate.

The modular group and congruence quotients

Trivial quotient: the unreduced hyperbolic plane

Choosing \(\Gamma=\{I\}\) removes quotient identifications: \[ \Gamma\backslash\mathbb H=\mathbb H. \] There is no fundamental-domain boundary and no exterior quotient tiling. The app still displays the determinant-\(\ell\) representatives used for the modular Hecke correspondence, but the resulting points remain at their actual locations in \(\mathbb H\) rather than being reduced into a quotient domain. Strictly speaking, this is not a Hecke operator intrinsic to the trivial quotient; it is an unreduced visualisation of the modular Hecke correspondence. See the Hecke / Bruhat–Tits tab for the double-coset distinction.

The modular group is \[ \mathrm{PSL}_2(\mathbb Z), \] acting by the same Möbius transformations as \(\mathrm{PSL}_2(\mathbb R)\). A standard fundamental domain is \[ \mathcal F = \left\{ z\in\mathbb H: |\operatorname{Re}z|\le\frac12,\ |z|\ge1 \right\}. \] Its vertical sides are paired by \(T:z\mapsto z+1\), and its circular sides by \(S:z\mapsto-1/z\).

For a prime level \(q\), the principal examples in the app are \[ \Gamma_0(q) = \left\{ \begin{pmatrix}a&b\\c&d\end{pmatrix} \in\mathrm{SL}_2(\mathbb Z): c\equiv0\pmod q \right\}, \] \[ \Gamma_1(q) = \left\{ \gamma\in\Gamma_0(q): a\equiv d\equiv1\pmod q \right\}, \] and \[ \Gamma(q) = \left\{ \gamma\in\mathrm{SL}_2(\mathbb Z): \gamma\equiv I\pmod q \right\}. \] In the geometric quotient, central signs act trivially, so one passes to the corresponding image in \(\mathrm{PSL}_2(\mathbb R)\).

Inclusions and indices at prime level

For a prime \(q\), \[ \Gamma(q)\subseteq\Gamma_1(q)\subseteq\Gamma_0(q)\subseteq\mathrm{SL}_2(\mathbb Z). \] In \(\mathrm{SL}_2(\mathbb Z)\) the indices are \[ [\mathrm{SL}_2(\mathbb Z):\Gamma_0(q)]=q+1, \] \[ [\mathrm{SL}_2(\mathbb Z):\Gamma_1(q)]=q^2-1, \] and \[ [\mathrm{SL}_2(\mathbb Z):\Gamma(q)] =q(q^2-1). \]

The geometry of the app is projective, so the relevant groups are their images in \(\mathrm{PSL}_2(\mathbb Z)\). Since \(-I\in\Gamma_0(q)\) for every \(q\), while \(-I\notin\Gamma_1(q),\Gamma(q)\) for odd \(q\), the effective projective indices for an odd prime \(q\) are \[ [\mathrm{PSL}_2(\mathbb Z):\overline{\Gamma_0(q)}]=q+1, \] \[ [\mathrm{PSL}_2(\mathbb Z):\overline{\Gamma_1(q)}] =\frac{q^2-1}{2}, \qquad [\mathrm{PSL}_2(\mathbb Z):\overline{\Gamma(q)}] =\frac{q(q^2-1)}{2}. \] At \(q=2\), the corresponding projective indices are \(3,3,6\).

Since the standard modular fundamental domain has hyperbolic area \[ \operatorname{area}\!\left(\mathrm{PSL}_2(\mathbb Z)\backslash\mathbb H\right) =\frac{\pi}{3}, \] a finite-index subgroup of projective index \(m\) has quotient area \[ \operatorname{area}(\Gamma\backslash\mathbb H)=m\,\frac{\pi}{3}. \] This is also why a congruence fundamental domain can be assembled from exactly \(m\) modular tiles up to boundary identifications.

These groups have finite index in the modular group. Their quotients are finite-area, generally noncompact hyperbolic orbifolds/surfaces with cusps. For noncompact AQUE, proving that limiting mass does not escape through the cusps is an additional issue.

Fundamental domains, side pairings, and tilings

If \(\Gamma<\mathrm{PSL}_2(\mathbb Z)\) has finite index and \(\{\gamma_j\}\) is a set of coset representatives, a fundamental domain for \(\Gamma\) can be assembled, up to boundary identifications, from finitely many translates \[ \gamma_j\mathcal F. \] This is the origin of the internal \(\mathrm{PSL}_2(\mathbb Z)\)-tiling visible inside a congruence-subgroup domain.

The exterior tiling applies elements of the chosen lattice to the selected fundamental domain. It therefore depicts many representatives of the same quotient geometry in the universal cover \(\mathbb H\) (or, after Cayley transform, in \(\mathbb D\)).

A fundamental-domain boundary is not a physical wall in the quotient. Paired sides are identified. When a trajectory crosses one side, reducing it back to the chosen domain replaces the representative by a lattice-equivalent point. The app uses this reduction to keep long trajectories numerically manageable without changing the underlying orbit.

Dirichlet domains

Around a base point \(p\), a Dirichlet domain has the form \[ \mathcal D_p = \{z:d(z,p)\le d(z,\gamma p)\ \text{for every }\gamma\in\Gamma\}. \] The compact quaternionic example uses this construction rather than a union of modular tiles.

Hecke correspondences and the Bruhat–Tits tree

At a good prime \(\ell\), the \(\ell\)-Hecke correspondence sends a generic point to \(\ell+1\) arithmetic neighbours. For the full modular group one convenient set of representatives is \[ z\longmapsto \ell z, \qquad z\longmapsto\frac{z+j}{\ell}, \quad j=0,\ldots,\ell-1. \] For a congruence quotient, a good prime satisfies \(\ell\nmid q\).

Before quotient projection, repeated nonbacktracking Hecke steps form the \((\ell+1)\)-regular Bruhat–Tits tree. The number of vertices at distance \(n\ge1\) from a root is \[ (\ell+1)\ell^{\,n-1}. \] The tree is an arithmetic object: tree distance measures \(\ell\)-adic separation, not hyperbolic distance.

The Hecke operator is, up to the chosen normalisation, the adjacency/neighbour-sum operator on this tree. This is why Hecke eigenfunctions and tree adjacency are directly related.

How the Hecke operator depends on the quotient

The clean general definition is by a double coset. For an arithmetic lattice \(\Gamma\) and a good prime \(p\), choose the local element represented by \[ a_p= \begin{pmatrix} p&0\\0&1 \end{pmatrix}. \] If \[ \Gamma a_p\Gamma = \bigsqcup_j \Gamma a_j, \] then, up to the chosen scalar normalisation, \[ (T_pf)(\Gamma g) = \sum_j f(\Gamma a_jg). \] At an unramified good prime there are \(p+1\) right cosets. Thus each point has \(p+1\) Hecke neighbours, and the corresponding local graph is the \((p+1)\)-regular Bruhat–Tits tree.

For the modular group one may take the explicit representatives \[ \begin{pmatrix}p&0\\0&1\end{pmatrix}, \qquad \begin{pmatrix}1&j\\0&p\end{pmatrix}, \quad j=0,\ldots,p-1, \] giving \[ z\longmapsto pz, \qquad z\longmapsto\frac{z+j}{p}. \] One common normalisation is therefore \[ (T_pf)(z) = \frac1{p+1} \left( f(pz)+\sum_{j=0}^{p-1}f\!\left(\frac{z+j}{p}\right) \right). \] Some sources instead use a factor such as \(1/\sqrt p\). This changes the normalisation of \(T_p\), not the underlying \(p+1\)-point Hecke correspondence.

For \(\Gamma_0(q)\), \(\Gamma_1(q)\), and \(\Gamma(q)\), the visualiser uses primes \(p\neq q\). At such a good prime the local \(p\)-adic subgroup is the same maximal compact subgroup, so the local spherical Hecke operator and the abstract \((p+1)\)-regular Bruhat–Tits tree are the same. What changes is the global quotient onto which the \(p+1\) arithmetic neighbours are projected: \[ \mathrm{SL}_2(\mathbb Z)\backslash\mathbb H,\qquad \Gamma_0(q)\backslash\mathbb H,\qquad \Gamma_1(q)\backslash\mathbb H,\qquad \Gamma(q)\backslash\mathbb H. \] Consequently the abstract tree can be identical while the visible Hecke points and quotient geodesics look very different.

At the level prime \(p=q\) this unramified picture changes. For example, on \(\Gamma_0(q)\backslash\mathbb H\) one encounters the \(U_q\) operator rather than the ordinary spherical \(T_q\). This is why the congruence examples in the visualiser restrict the selectable Hecke prime to \(p\neq q\).

For the compact quaternionic quotient the global arithmetic realisation is different, but the good-prime local picture is again the same. At a split, unramified prime, \[ D\otimes_{\mathbb Q}\mathbb Q_p \cong M_2(\mathbb Q_p), \] and the local correspondence again gives a \((p+1)\)-regular Bruhat–Tits tree. Globally, however, the neighbours are realised using quaternionic elements or ideals of reduced norm \(p\), rather than the elementary modular matrices above.

Important caveat: the trivial quotient does not have this \(p+1\)-branch Hecke operator

When the app selects \[ \Gamma=\{I\}, \qquad \Gamma\backslash\mathbb H=\mathbb H, \] the displayed maps \[ z\mapsto pz, \qquad z\mapsto\frac{z+j}{p} \] are not a Hecke operator intrinsically attached to the unquotiented hyperbolic plane.

If the double-coset definition is applied literally to the trivial group, then \[ \Gamma a_p\Gamma=\{a_p\}, \] so there is only one double-coset branch rather than the usual \(p+1\) Hecke neighbours. The app instead keeps the ordinary modular determinant-\(p\) representatives and simply omits quotient reduction.

Trivial-quotient mode should therefore be interpreted as showing the unreduced representatives of the modular \(p\)-Hecke correspondence. It is useful for seeing how the arithmetic points sit in the universal cover before a lattice identifies or reduces them, but it should not be called “the Hecke operator on \(\mathbb H\)”.

The useful hierarchy is \[ \text{good-prime local Hecke geometry} \;+\; \text{global arithmetic quotient} \;\longrightarrow\; \text{projected Hecke correspondence}. \] The genuine arithmetic quotients in the app share the same good-prime local Bruhat–Tits geometry; the trivial mode is included only to separate this arithmetic correspondence from the geometric effect of quotient reduction.

The app's Arithmetic Hecke walk is only a visual random walk on these adjacency relations: at each step the root chooses uniformly among its \(\ell+1\) current neighbours. The chosen abstract edge is highlighted simultaneously in the Bruhat–Tits diagram and in the quotient pictures.

Projection to the quotient

Distinct tree vertices may project to the same visible quotient point. The app therefore keeps symbolic Hecke vertices distinct and separately computes the shortest quotient geodesic between projected endpoints: \[ d_{\Gamma\backslash\mathbb H}([z],[w]) = \inf_{\gamma\in\Gamma}d_{\mathbb H}(z,\gamma w). \] The Bruhat–Tits diagram and hyperbolic plot share the same generation styles.

Hecke recurrence

Hecke recurrence is a property of a measure on the quotient, not a claim that a random walk on the abstract tree returns to its initial tree vertex. A useful formulation is: if \(B\) is measurable with \(\mu(B)>0\), then for \(\mu\)-almost every \(x\in B\) and every finite Hecke radius \(R\), there is a point \(y\in B\) in the Hecke orbit of \(x\) with \[ d_{\ell}(x,y)>R. \] Thus points can return geometrically to the same positive-measure set while escaping arbitrarily far in the arithmetic/Bruhat–Tits direction.

The app's compact quaternionic arithmetic model

The implemented example uses \[ D=(7,-1)_{\mathbb Q}, \qquad i^2=7,\quad j^2=-1,\quad ij=-ji, \] represented in \(M_2(\mathbb R)\) by \[ a+bi+cj+dij \longmapsto \begin{pmatrix} a+\sqrt7\,b & c+\sqrt7\,d\\ -c+\sqrt7\,d & a-\sqrt7\,b \end{pmatrix}. \] Its reduced norm is \[ \operatorname{Nrd}(a+bi+cj+dij) = a^2-7b^2+c^2-7d^2. \]

The app uses the order \[ \mathcal O=\mathbb Z[1,i,j,ij] \] and its norm-one units, modulo the central sign, to obtain a lattice in \(\mathrm{PSL}_2(\mathbb R)\). Because the underlying quaternion algebra is a division algebra over \(\mathbb Q\) but splits over \(\mathbb R\), the corresponding arithmetic quotient is compact: there are no cusps and no escape of mass.

The displayed Dirichlet polygon is centred at \(2i\). At good primes \(\ell\ne2,7\), the local arithmetic again gives an \((\ell+1)\)-regular Bruhat–Tits tree. The visualiser treats the polygon numerically; it should not be interpreted as an interval-arithmetic certification of the exact Dirichlet cell.

AQUE, microlocal lifts, Bowen balls, and entropy

If \(\phi_j\) is a high-energy Laplace eigenfunction, its configuration-space probability density \(|\phi_j|^2\,dA\) remembers position but not direction. A microlocal lift records observables on phase space: \[ \mu_j(a) = \langle \operatorname{Op}(a)\phi_j,\phi_j\rangle, \qquad a\in C_c^\infty(T^1X) \] (schematically; the precise quantisation and normalisation depend on convention). Weak-* limits of these distributions/measures are quantum limits.

What QUE asserts

Quantum unique ergodicity asserts that every microlocal quantum limit is the normalised Liouville/Haar measure on \[ T^1X\cong\Gamma\backslash\mathrm{PSL}_2(\mathbb R), \] rather than merely saying this for a density-one subsequence as in quantum ergodicity. Projecting the microlocal measure to the base surface recovers the configuration-space measure: for an observable \(f=f(x)\) depending only on position, \[ \mu_j(f)=\int_X f(x)|\phi_j(x)|^2\,dA(x). \] Consequently, QUE for the microlocal lifts implies equidistribution of the original \(|\phi_j|^2dA\) measures on \(X\).

Egorov/commutator arguments imply that quantum limits are invariant under geodesic flow \(A\). In the arithmetic setting, simultaneous Hecke symmetry supplies additional recurrence in the \(\ell\)-adic/Hecke direction.

Bowen balls

For the time-one geodesic map \(a=a_1\), an \(n\)-step Bowen ball is \[ B_n(x,\delta) = \{y:d(a^k x,a^k y)<\delta\ \text{for all }|k|\le n\}. \] Locally, the stable and unstable widths of such a set are exponentially small in \(n\). Exponential decay of \(\mu(B_n(x,\delta))\) gives positive Kolmogorov–Sinai entropy.

Bourgain–Lindenstrauss local product box

The panel uses \[ B(\varepsilon,\tau) = A_{(-\tau,\tau)} U^-_{(-\varepsilon,\varepsilon)} U^+_{(-\varepsilon,\varepsilon)}. \] In the BL projection panel the \(A\)-parameter is written with \[ a(t)=\operatorname{diag}(e^t,e^{-t}), \] so acting on \(i\) after \(a(t)u^-(r)u^+(s)\) gives \[ y=\frac{e^{2t}}{1+s^2}, \qquad x=e^{2t}\left(r+\frac{s}{1+s^2}\right). \] This is a local-product neighbourhood comparable to a Bowen ball after the transverse scale \(\varepsilon\) is chosen exponentially small in the orbit length.

The arithmetic QUE strategy represented by the app is therefore \[ \text{microlocal lift} \longrightarrow A\text{-invariant quantum limit} \quad+\quad \text{Hecke recurrence} \quad+\quad \text{positive entropy} \longrightarrow \text{measure rigidity} \longrightarrow \text{Haar/Liouville measure}. \] In a noncompact quotient one also needs to prevent escape of mass into the cusps; in the compact quaternionic model that issue is absent.

How to read the visualiser

The red central point represents the base point of the current frame. Its tangent direction is stored separately. During \(K=\mathrm{SO}(2)\) rotation the red base point therefore stays fixed while the tangent angle changes.

The half-plane and disk canvases are related exactly by the Cayley transform described in the Hyperbolic models tab. They are not independent simulations. During continuous flow, every displayed projected Hecke point is updated on the same rendered movement frame as the central root, rather than using a deliberately slower Hecke refresh clock. Panning or changing coordinates changes the display, not the underlying quotient object.

Internal modular tiling shows the copies of the standard modular domain used to assemble a congruence fundamental domain. When both tilings are enabled, this modular subdivision is shown only inside the currently selected congruence fundamental domain, while the exterior is tiled by whole copies of the larger selected \(\Gamma\)-fundamental domain. The exterior cache extends beyond the visible window and the renderer discards only copies that are clearly off-screen or already sub-pixel, so the tiling can approach the ideal boundary more closely without drawing every remote lattice translate. In colour-fill modes, distinct tiles are deliberately assigned distinct deterministic colours so their cell structure can be read immediately; regions not covered by a displayed tile retain the ordinary black or white plot background. The colours are only a visualisation of the cells: the quotient geometry is determined by the side identifications, not by the fill colours.

The four panel buttons at the top are independent visibility toggles: half-plane, disk, Bruhat–Tits tree, and density visualiser may be displayed simultaneously. On a wide page the four panels form one row, then a \(2\times2\) grid, and finally a single column as the page narrows. Pause the dynamics and hover a visible component to edit its appearance. The Bruhat–Tits diagram and hyperbolic Hecke tree use the same style keys: changing a generation of points or edges in either view changes both. Boundary-pair colours/textures may differ by identified pair, while fundamental-domain boundary thickness is shared globally.

Top-panel interaction

The half-plane, disk, Bruhat–Tits tree, and density visualiser are independent display panels. The first three are shown by default; the density visualiser can be added with its panel button. On a wide page the visible panels share a row, then reorganise into a \(2\times2\) grid and finally a single column as the page narrows.

The Bruhat–Tits tree has no separate styling system: when paused, hovering a node or edge opens the same editor used by the projected Hecke tree on the hyperbolic surface, and edits in either location update both. The density visualiser's controls are intentionally kept below the image; its empirical mode records finite-time orbit positions, while its reference mode shows normalised hyperbolic-area density.

A recurrence region \(B\) can be drawn from either hyperbolic model. Drawing in the disk is converted by the inverse Cayley transform to the same underlying region in \(\mathbb H\), so the recurrence set displayed in the disk and half-plane represents one shared mathematical object.

Global Monochrome overrides all stored custom colours without deleting them. Returning to Colour restores the saved colours. The exact mathematical geometry is independent of these display choices.

ARITHMETIC HYPERBOLIC DYNAMICS VISUALISER

Mathematical notes and interpretation