Spectral dynamics for the Basilica group

Abstract: The projective spectrum of a tuple $A=(A_0,A_1,\dots,A_n)$ with elements in a unital Banach algebra $\mathcal{A}$ is the collection of $[z_0:z_1:\cdots:z_n]\in \mathbb{P}<sup>n$ such that $z_0A_0+z_1A_1+\dots+z_nA_n$ is not invertible in $\mathcal{A}$. For the tuple $A_ρ=(ρ({\bf e}),ρ({\bf a}),ρ({\bf b}))$, where ${\bf e},{\bf a},{\bf b}$ are the three states of the automaton generating the Basilica group $\mathcal{B}$ and $ρ$ is the Koopman representation, a dynamical map $F$ preserving the projective spectrum of $A_ρ$ is established. Further, a substantial common subset of the Julia set of $F$ and the projective spectrum of $A_ρ$, as well as a large common subset of the Fatou set of $F$ and the projective resolvent set of $A_ρ$ is identified. The result provides partial support for the conjecture of equality of the Julia set of $F$ and the projective spectrum of $A_ρ$.

• The paper extends spectral dynamics to the Basilica group, constructing an explicit dynamical map on $\mathbb{P}^2$ that preserves both the projective spectrum and its complement.
• The authors identify substantial common subsets of the Julia set and the projective spectrum, partially supporting a conjecture about their full coincidence.
• The work provides the strongest form of agreement between spectral and dynamical sets, characterized explicitly on coordinate hyperplanes.

This paper by Hongyi Liu and Wei He extends the program of spectral dynamics—initiated for the infinite dihedral group and the lamplighter group—to a new self-similar group, the Basilica group $\mathcal{B}$. The central object is the projective spectrum of the operator tuple $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$, where $\rho$ is the Koopman representation of $\mathcal{B}$ on $L^2(\partial T,\mu)$ for the uniform Bernoulli measure on the boundary of the rooted binary tree. The authors construct an explicit dynamical map $F$ on $\mathbb{P}^2$ that preserves both the projective spectrum and its complement, and they identify substantial common subsets of $\mathcal{J}(F)$ with $p(A_\rho)$ and of $\mathcal{F}(F)$ with $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$0. The work provides partial support for the conjecture that these sets coincide in full.

The dynamical map from self-similarity

The Basilica group is generated by a $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$1-automaton with states $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$2 acting on the binary tree via the wreath recursions

$$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$3

Under the Koopman representation, self-similarity yields a unitary $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$4 conjugating the generators into block matrices. A Schur-complement argument then shows that $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$5 is invertible if and only if $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$6 is invertible, where

$$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$7

Projectivizing gives the rational map on $$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$8,

$$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$$9

defined off its extended indeterminacy set $\rho$0. The indeterminacy set is computed explicitly: starting from $\rho$1 and iterating preimages, one obtains points $\rho$2 and $\rho$3 with $\rho$4, whose closure over all $\rho$5 is the unit circle. Hence

$\rho$6

A notable corollary follows immediately: since $\rho$7, the spectra of both $\rho$8 and $\rho$9 are exactly the unit circle—a clean spectral fact derived purely from the dynamics.

Fatou set and projective resolvent set

Let $\mathcal{B}$0 for $\mathcal{B}$1. The first main theorem states

$\mathcal{B}$2

The resolvent inclusion is elementary: on each $\mathcal{B}$3, dominance of one coordinate makes $\mathcal{B}$4 a norm-perturbation of an invertible operator (using unitarity of $\mathcal{B}$5, $\mathcal{B}$6, and $\mathcal{B}$7). The Fatou inclusion requires more care. On $\mathcal{B}$8, the map takes the normal form $\mathcal{B}$9, and the recursive coordinates satisfy estimates showing both odd and even subsequences converge uniformly to $L^2(\partial T,\mu)$0 on neighborhoods within $L^2(\partial T,\mu)$1. On $L^2(\partial T,\mu)$2, orbits either escape to the attracting 2-cycle $L^2(\partial T,\mu)$3 or hit the hypersurface $L^2(\partial T,\mu)$4 in finite time; uniform convergence of even and odd iterates is established in both regimes, with convergence rates independent of the hitting time. The region $L^2(\partial T,\mu)$5 maps into $L^2(\partial T,\mu)$6 under $L^2(\partial T,\mu)$7, reducing it to the previous case.

Julia set and projective spectrum

The second main theorem identifies a large spectral subset inside the Julia set:

$L^2(\partial T,\mu)$8

Spectral side. The finite-level representations $L^2(\partial T,\mu)$9 on $F$0 satisfy the same matrix recursion, so $F$1. Since $F$2, this gives $F$3, and taking the union over $F$4 and closing yields the inclusion into $F$5.

Dynamical side. The delicate part is proving $F$6. For points $F$7 with all coordinates nonzero, $F$8 swaps $F$9 with $\mathbb{P}^2$0, so $\mathbb{P}^2$1 fixes $\mathbb{P}^2$2. Assuming $\mathbb{P}^2$3 were Fatou, Arzelà–Ascoli gives equicontinuity of $\mathbb{P}^2$4 in local coordinates. The Jacobian of $\mathbb{P}^2$5 at the fixed point is

$\mathbb{P}^2$6

with eigenvalues $\mathbb{P}^2$7 and $\mathbb{P}^2$8. The repelling eigenvalue forces expansion along eigenvectors: using a cone invariant under $\mathbb{P}^2$9 (splitting $\mathcal{J}(F)$0 with $\mathcal{J}(F)$1), the authors show $\mathcal{J}(F)$2 while remaining bounded—an impossibility. Thus every such point lies in $\mathcal{J}(F)$3. Points of $\mathcal{J}(F)$4 with zero coordinates are handled separately: those with $\mathcal{J}(F)$5 or $\mathcal{J}(F)$6 lie in $\mathcal{J}(F)$7, and $\mathcal{J}(F)$8 is shown to be non-Fatou by a sequence argument exhibiting discontinuity of any putative uniform limit.

An important consequence is that the two correspondences are exact on coordinate hyperplanes: for each $\mathcal{J}(F)$9,

$p(A_\rho)$0

with the common set characterized explicitly (e.g., $p(A_\rho)$1), and dually for the resolvent/Fatou pair. This is the strongest form of agreement established in the paper.

Limitations and open questions

The paper does not establish equality between the Julia set and the full projective spectrum; only the inclusions above are proved, and the authors state plainly that whether the sets coincide "is not yet known." Two questions are posed:

• Are $p(A_\rho)$2, $p(A_\rho)$3, and $p(A_\rho)$4 pairwise equal?
• Are $p(A_\rho)$5, $p(A_\rho)$6, and $p(A_\rho)$7 pairwise equal?

Numerical simulations suggest affirmative answers, but no proof is given. The authors also observe an asymmetry in difficulty: verifying membership in the projective spectrum is comparatively easy, whereas establishing the corresponding dynamical statement (Julia or Fatou membership) is considerably harder—suggesting the spectrum can serve as a computable proxy for the Julia set. Complete details are deferred to a separate paper.

Conclusion

The paper adds the Basilica group to the short list of self-similar groups admitting a computable spectral dynamics, deriving the map $p(A_\rho)$8 directly from the wreath recursion structure. Its main contributions are the explicit computation of the extended indeterminacy set, the identification of large common subsets of $p(A_\rho)$9 and $\mathcal{F}(F)$0, and the exact coincidence of the pairs on coordinate hyperplanes. Whether the identified subsets exhaust the intersections remains open, and resolving this would extend the complete correspondence known for the infinite dihedral group to a group of exponential growth.