Operators of stochastic adding machines and Julia sets

Ali Messaoudi; Danilo Caprio; Glauco Valle; Ioannis Tsokanos

REVIEW 3 major objections 6 minor 22 references

Operators of stochastic adding machines and Julia sets

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a bounded Cantor numeration system, the transition operator of a stochastic adding machine has spectrum equal to a fibered filled Julia set in the null-recurrent case and to its boundary in the transient case, with eigenvalues exactly…

desk verdict The null-recurrent half and the point-spectrum side are solid and new; the transient half of Theorem 4.1 has a genuine, load-bearing gap at (5.21), so the paper needs referee attention rather than desk rejection. read the letter →

arxiv 2506.05703 v1 pith:6CRR4AWN submitted 2025-06-06 math.DS

Danilo Caprio , Ali Messaoudi , Ioannis Tsokanos , Glauco Valle This is my paper

classification math.DS MSC 37F1047A1060J05

keywords stochasticaddingmachineCantornumerationsystemfiberedfilledJuliasettransitionoperatorspectrumMarkovchaincompactification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the stochastic adding machine—a Markov chain that adds one to a number by updating digits one at a time, with random failures that can stop the procedure—from the non-negative integers to their compactification under a Cantor numeration system. The main theorem says that for a bounded digit sequence the spectrum of the induced transition operator on continuous functions is exactly a fibered filled Julia set $\mathcal{E}$ when the infinite product of success probabilities is zero, and exactly the boundary $\partial\mathcal{E}$ when that product is positive. In both cases the eigenvalues are precisely the points that some iterate of the digit polynomials sends to $1$, a countable set whose closure is $\partial\mathcal{E}$. This is the natural bridge between a probabilistic dichotomy—null recurrence versus transience of the machine—and the complex-dynamical dichotomy between a compact set and its boundary.

What carries the argument

The load-bearing object is the fibered filled Julia set $\mathcal{E}_{\bar d,\bar p}$, the set of $z\in\mathbb{C}$ whose forward orbit under the iterates $\tilde f_r=f_r\circ\cdots\circ f_1$ remains bounded, where each digit step is the polynomial $f_r(z)=((z-(1-p_r))/p_r)^{d_r}$. Two pieces of machinery carry the argument: Lemma 5.2, which identifies $\mathcal{E}_{\bar d,\bar p}$ with the eigenvalue set of the original discrete operator and with the set of $\lambda$ for which the coordinate sequence $(\iota_\lambda(r))_{r\ge 1}$ stays bounded; and Lemma 5.3, which equates $f_r(\sigma(\tilde S_r))$ with $\sigma(\tilde S_{r+1})$ through the embedding operators $\tilde\Pi_{k,r}$ and the spectral mapping theorem. In the transient case the governing mechanism is Proposition 5.6: spectral points $\lambda$ must satisfy $\lim_{r\to\infty}|\tilde f_r(\lambda)|=1$, so the spectrum cannot enter the interior of $\mathcal{E}$.

What would settle it

Take a bounded digit sequence such as $d_r=3$ for all $r$, with $p_1=1/2$ and $p_r=1-2^{-r}$ for $r\ge 2$, so that $\prod_r p_r\approx 0.29\in(0,1/2)$. Compute the spectrum of $\tilde S_{\bar d,\bar p}$ on $C(\Gamma_{\bar d})$; if any point of the interior of $\mathcal{E}_{\bar d,\bar p}$ lies in the spectrum, or if some $\lambda$ in the spectrum has $\limsup_{r\to\infty}|\tilde f_r(\lambda)|\ne 1$, then the transient case of Theorem 4.1 fails as stated.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.1. For bounded $\bar d=(d_r)_{r\ge 1}$ with $d_r\ge 2$ and $\bar p=(p_r)_{r\ge 1}$ with $p_r\in(0,1]$, the operator $\tilde S_{\bar d,\bar p}$ acting on $C(\Gamma_{\bar d})$ satisfies $\sigma(\tilde S_{\bar d,\bar p})=\sigma_{\mathrm{ap}}(\tilde S_{\bar d,\bar p})=\mathcal{E}_{\bar d,\bar p}$ when $\prod_{r\ge 1}p_r=0$, and $=\partial\mathcal{E}_{\bar d,\bar p}$ when $\prod_{r\ge 1}p_r>0$. The point spectrum is the union $\bigcup_{r\ge 1}\tilde f_r^{-1}\{1\}$, and that union is dense in $\partial\mathcal{E}_{\bar d,\bar p}$. The proof obtains the inclusions through a spectral mapping relation linking $f_r$ applied to the spectrum of the $r$-th shifted operator to the spectrum of the next one, and identifies the boundary dynamically via Montel's theorem on normal families.

Load-bearing premise

The transient-case part of the proof rests on Proposition 5.6, whose written argument assumes the product of success probabilities is greater than $1/2$, even though the theorem states the boundary result for every positive product; if that restriction cannot be removed, the claim for products in $(0,1/2]$ is unsupported.

Editorial extensions

If this is right

The spectral radius and spectrum of the compactified operator are determined by the digit sequence and the single number $\prod_r p_r$; no finer information about the failure probabilities is needed.
When $\prod_r p_r=0$, every point of the filled Julia set is an approximate eigenvalue, so the approximate point spectrum covers the whole compact set.
When $\prod_r p_r>0$, the spectrum shrinks to the boundary, so the residual and continuous spectra must arrange themselves so that the approximate spectrum equals that boundary.
The eigenvalue set, being countable and dense in $\partial\mathcal{E}$, gives an explicit spectral decomposition piece in terms of preimages of the point $1$.
When all success probabilities are $1$, the operator acts as the shift $g(x)\mapsto g(x+1)$ and the spectrum collapses to the unit circle, matching the model case treated in the proof.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether boundedness of $\bar d$ is needed; one natural test is whether the density of $\bigcup \tilde f_r^{-1}\{1\}$ in $\partial\mathcal{E}$ persists for slowly growing digits such as $d_r=r+1$.
The written proof of Proposition 5.6 assumes $\prod_r p_r>1/2$; until that restriction is removed or reduced, the transient case of Theorem 4.1 should be read as proved for products above $1/2$ and conjectured for the remaining positive products.
A numerical experiment on the boundary claim could look at $d_r=3$ with $p_1=1/2$, $p_r=1-2^{-r}$ for $r\ge 2$: the theorem predicts the spectrum is $\partial\mathcal{E}$, and finding an interior spectral point would falsify it.
The identification of eigenvalues with preimages of $1$ suggests that return-time or mixing statistics of the Markov chain on the compactified space may be expressible through the iterated preimage tree of the point $1$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Your read is close to mine. The genuinely new thing here is the compactification: extending the stochastic adding machine to Gamma_d and computing the spectrum of the induced operator on C(Gamma_d). The E-vs-boundary dichotomy according to whether the product of the p_r is zero is a clean structural statement, and the null-recurrent inclusion E subset sigma_ap is argued in detail and looks sound. The unconditional inclusion sigma subset E via Lemma 5.3 is also solid, and the point-spectrum identification with the union of the preimages of 1, plus its density in the boundary, is a nice piece of work. The reliance on Lemma 5.2 from [19] is transparent, not circular.

The soft spots are exactly where you put them. Proposition 5.6 is proved only under the extra assumption that the infinite product exceeds 1/2. I suspect this is a routine tail-shift fix: once the tail product exceeds 1/2, apply the shifted operator. But the written proof does not say that. The more serious problem is (5.21). Equation (4.3) gives only that the boundary is the closure of the union of preimages of 1; it gives density, not containment. From that, asserting that for every component W of int(E) one has 1 in the boundary of f_r(W) for all large r is a non-sequitur. The subsequent angle-set and doubling argument needs intervals (0, epsilon_r) in the angle sets, so without a real proof of (5.21) the claim that no interior point of E lies in the spectrum is unsupported. This is not a cosmetic gap. Remark 5.2's easier route only handles connected E, which is not the general setting.

Net: I would send it to a serious referee rather than desk reject. The null-recurrent half and the general spectral inclusions are worth publishing on their own, and the transient theorem is plausible and probably repairable. But the referee should require the authors to either prove (5.21) or replace the transient argument. As it stands, I would not rely on the sigma = boundary half.

Referee Report

3 major / 6 minor

Summary. The paper studies stochastic adding machines over bounded Cantor numeration systems and extends the Markov chain from Z_+ to the compact space Γ_d. It defines a transition operator S-tilde on C(Γ_d) and characterizes its spectrum in terms of the fibered filled Julia set E_{d,p}. Theorem 4.1 claims that for bounded d, the spectrum and approximate point spectrum coincide with E_{d,p} when the infinite product of the success probabilities is zero, and with ∂E_{d,p} when the product is positive. It also claims that the point spectrum is exactly the union of the preimages of 1 under the iterated polynomials, and that this union is dense in the boundary. The proof proceeds through an unconditional spectral inclusion, a null-recurrent constructive argument, and a transient-case dynamical argument using normal families and angle estimates.

Significance. If fully established, the theorem is a natural and nontrivial extension of the Killeen-Taylor result and of the spectrum results in [19] from l∞(Z_+) to the continuous function space on the compactified state space. The paper gives detailed arguments for the unconditional spectral inclusion and for the null-recurrent equality, and it provides explicit eigenfunctions as well as a Montel-theorem proof that the preimages of 1 are dense in the boundary. The examples and figures are helpful. However, the transient-case proof has load-bearing gaps that must be repaired before the stated theorem is supported.

major comments (3)

[§5.2.2, relation (5.21)] The proof of (5.13) asserts: 'Since ∂W ⊆ ⋃_r f_r^{-1}{1}, one may, without loss of generality, choose r0 ... such that ... 1 ∈ ∂ f_r(W) for all r ≥ r0−1.' This does not follow from (4.3), which only gives ∂E = closure of ⋃_r f_r^{-1}{1}; therefore ∂W is contained in the closure, not in the union. Even if some boundary point z of W were a preimage of 1 at some index, density does not imply that 1 lies on the boundary of every f_r(W) for all large r. The subsequent angle-set construction, including the assertions that A_r contains (0,ε_r), the doubling estimate a_{r+1}≥2a_r, and the conclusion f_{r_w+1}(λ_w)∈[0,1), depends on (5.21). Without (5.21), the dichotomy (5.18) cannot be forced to the {0} alternative, so (5.20), (5.19), (5.13), and ultimately the inclusion σ⊆∂E in the transient case are unsupported.
[§5.2.2, Proposition 5.6] Proposition 5.6 is stated for every sequence with ∏ p_r>0, but its proof begins 'Assume that ∏ p_r >1/2' and gives no reduction to the general product-positive case. The lower bound in (5.27) uses the constant C=∏ p_r^{-1}<2, which is unavailable when 0<∏ p_r≤1/2. Since Proposition 5.6 is the step that forces |f_r(λ)|→1 for λ∈σ and thereby rules out interior spectral points, the transient-case proof is incomplete in the generality stated in Theorem 4.1.
[§5.2.2, near (5.24)] When (2π−ε0,2π)⊆A_{r0}, the text says that 'we may instead consider the complex conjugates λ of λ∈W' and then assumes (0,ε_{r0})⊆A_{r0}. This reduction needs justification because W need not be conjugation-invariant. Since the polynomials have real coefficients, E is conjugation-invariant, so one could apply the argument to the conjugate component and conjugate back; this step, however, is not written and is needed for the existence of λ_w∈W.

minor comments (6)

[Abstract and Section 1] The phrase 'and/of psychological judgment' should read 'and/or psychological judgment'.
[Section 7, proof of Lemma 5.2] The inclusion σpt(S_{d,p}, l∞(Z+))⊆E_{d,p} is imported from [19, Lemma 3.6] rather than proved in the Annex; this dependency should be stated explicitly in the lemma statement.
[§5.2.1, proof of (4.2)] The sentence 'gλ(n)=c·vλ(n) for all n∈Γ_d' should say 'for all n∈Z+' (or for all n in the dense copy of Z+ in Γ_d), since vλ is defined as a sequence on Z+.
[§5.2.2, relation (5.13)] The notation 'inter(E_{d,p})' should be 'int(E_{d,p})' or 'the interior of E_{d,p}' for consistency with the rest of the paper.
[References] Reference [13] lists pages '1998–1903'; this is likely a typo and should be corrected to the correct page range.
[§5.2.2, Remark 5.2] Remark 5.2 offers a simpler proof only under an additional connectedness assumption on f_{r0}(E); it should be clear that this is an extra hypothesis and not a substitute for the general proof of (5.13).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fibered filled Julia set is defined independently and the spectral result is not fitted to it; the one self-citation is a transparent import of a published lemma, not a circular reduction.

full rationale

The central object E is defined in (1.7) by the boundedness of the iterates f-tilde_r, with f_r given in (1.8); Theorem 4.1 then asserts an equality of spectra with E or its boundary. No parameter in E is fitted to the spectrum and no eigenvalue is defined in terms of E; the statement is therefore not self-definitional. The proof uses Lemma 5.2, which identifies the point spectrum of the original l-infinity operator with E; a more detailed statement is proved in Section 7, with the inverse eigenvector characterization imported from the authors' own [19, Lemma 3.6]. This is the only noticeable self-citation, and it is not circular in the prohibited sense: [19] is a published, parameter-free theorem with stated assumptions that do not include the present result, and the new contribution concerns the extended operator on C(Gamma_dbar), whose point spectrum (union of preimages of 1, dense in the boundary) is genuinely different from the old l-infinity spectrum. The remaining chain - Proposition 5.1, Lemma 5.3, Lemma 5.4, Montel arguments, and the constructive approximate-spectrum proof in the null-recurrent case - does not reduce its conclusion to its inputs. Two genuine proof gaps exist in the transient case (Section 5.2.2): Proposition 5.6 is proved only under the extra assumption that the infinite product of p_r exceeds 1/2, without a written reduction to the general positive-product case, and the inference to (5.21) ('1 in boundary of f-tilde_r(W) for all large r') is not justified by the density statement (4.3). These are correctness risks, not circularity, because they do not make the theorem equivalent to an assumption or to a fitted quantity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

There are no fitted constants or ad hoc parameters: the digit sequence (d_r) and the success probabilities (p_r) are structural inputs to the model. The paper relies on standard theorems, on Lemma 5.2 imported from [19], and on the explicit boundedness assumption on d. No new physical or mathematical entities are postulated beyond the already standard compactification Gamma_d.

assumptions (4)

domain assumption The eigenvector characterization of S on l-infinity(Z+): every eigenvalue lies in E and every eigenvector is c times v-lambda (Lemma 5.2, citing [19, Lemma 3.6]).
Imported from the authors' earlier paper [19]; used to identify eigenvalues of the extended operator S-tilde and to control the sequences iota-lambda(r).
standard math Standard spectral theory facts: the inclusions sigma_pt subset sigma_ap subset sigma and the spectral mapping theorem for polynomials.
Used in Section 2 and in Lemma 5.3 to shift spectra through the polynomial maps f_r.
standard math Montel's theorem on normal families, including the omitted-values criterion.
Used in Lemma 5.4 and in the proof of the dichotomy in (5.18).
domain assumption The sequence d is bounded.
Explicit assumption of Theorem 4.1; needed for Claim 5.5 and Lemma 5.4(2), and for the transient-case spectral inclusion.

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Pith. "Pith review of Operators of stochastic adding machines and Julia sets." pith.science (2026). https://pith.science/paper/6CRR4AWN

@misc{pith2026250605703,
  author       = {Pith},
  title        = {Pith review of: Operators of stochastic adding machines and Julia sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CRR4AWN}},
  note         = {Machine review of arXiv:2506.05703}
}

abstract

A stochastic adding machine is a Markov chain on the set of non-negative integers $\mathbb{Z}_{+}$ that models the process of adding one by successively updating the digits of a number's expansion in a given numeration system. At each step, random failures may occur, interrupting the procedure and preventing it from continuing beyond a certain point. The first model of such a stochastic adding machine, constructed for the binary base, was introduced by Killeen and Taylor. Their work was motivated by applications to biological clocks, aiming to model phenomena related to time discrimination and/of psychological judgment. From a mathematical perspective, they characterized the spectrum of the associated transition operator in terms of a filled Julia set. In this paper, we consider a stochastic adding machine based on a bounded Cantor numeration system and extend its definition to a continuous state space--namely, the closure of $\mathbb{Z}_+$ with respect to the topology induced by the Cantor numeration system. This stochastic process naturally induces a transition operator $S$ acting on the Banach space of continuous complex-valued functions over the continuous state space, as well as a fibered filled Julia set $\mathcal{E}$. Our main result describes the spectrum of $S$ in terms of the fibered filled Julia set $\mathcal{E}$. Specifically, if the stochastic adding machines halts with probability one after a finite number of steps, then the spectrum of $S$ coincides with $\mathcal{E}$; otherwise, the spectrum coincides with the boundary $\partial \mathcal{E}$.

Figures

Figures reproduced from arXiv: 2506.05703 by the authors.

**Figure 1.1.** Initial parts of the transition graph of AMFC [PITH_FULL_IMAGE:figures/full_fig_p004_1_1.png] view at source ↗

**Figure 5.2.** Illustration of the trigonometric relations (5.22) and (5.23). [PITH_FULL_IMAGE:figures/full_fig_p018_5_2.png] view at source ↗

Figures from the paper (8 more)

**Figure 8.3.** Figure 8.3: The set Ed,¯p¯ for the cases a) p1 = 0.7 and pr = 1 for r ≥ 2, b) p1 = 0.5 and pr = 1 for r ≥ 2, c) p1 = 0.4 and pr = 1 for r ≥ 2. a) b) c) [PITH_FULL_IMAGE:figures/full_fig_p028_8_3.png]

**Figure 8.4.** Figure 8.4: The set Ed,¯p¯ for the cases a) p1 = p2 = p3 = 0.8 and pr = 1 for r ≥ 4, b) p1 = p2 = p3 = 0.7 and pr = 1 for r ≥ 4, c) p1 = p2 = p3 = 0.6 and pr = 1 for r ≥ 4. a) b) c) [PITH_FULL_IMAGE:figures/full_fig_p028_8_4.png]

**Figure 8.5.** Figure 8.5: The set Ed,¯p¯ for the cases a) p2 = 1, p3 = 0.5 and pr = 0.55 for r ̸= 2, 3, b) p2 = 1 and pr = 0.55 for r ̸= 2, c) p1 = 1 and pr = 0.55 for r ̸= 1. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_8_5.png]

**Figure 8.6.** Figure 8.6: The set Ed,¯p¯ for the cases a) pr = 0.8, b) pr = 0.6, c) pr = 0.52, for all r ≥ 1. a) b) c) [PITH_FULL_IMAGE:figures/full_fig_p029_8_6.png]

**Figure 8.7.** Figure 8.7: The set Ed,¯p¯ for the cases a) p2 = 1, p3 = 0.5 and pr = 0.55 for r ̸= 2, 3, b) p2 = 1 and pr = 0.55 for r ̸= 2, c) p1 = 1 and pr = 0.55 for r ≥ 2. a) b) c) [PITH_FULL_IMAGE:figures/full_fig_p029_8_7.png]

**Figure 8.8.** Figure 8.8: The set Ed,¯p¯ for the cases a) pr = 0.55, b) pr = 0.81, c) pr = 0.61, for all r ≥ 1. a) b) c) [PITH_FULL_IMAGE:figures/full_fig_p029_8_8.png]

**Figure 8.9.** Figure 8.9: The set Ed,¯p¯ for the cases a) p2 = 0.9 and pr = 0.55 for r ̸= 2, b) p2 = 1 and pr = 0.695 for r ̸= 2, c) p1 = 0.55, p2 = p3 = p4 = 0.95 and pr = 0.55 for r ̸= 5. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_8_9.png]

**Figure 8.10.** Figure 8.10: The set Ed,¯p¯ for the cases a) pr = 0.7, b) pr = 0.704, c) pr = 0.8, for all r ≥ 1. Acknowledgment Ali Messaoudi was partially supported by CNPq grant 310784/2021-2 and by Fapesp grant 2019/10269-3. Ioannis Tsokanos was supported by Fapesp grant 2024/10135-5. Glauc…

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Works this paper leans on

22 extracted references · 22 canonical work pages

[19]

Messaoudi, G

A. Messaoudi, G. Valle, Spectra of generalized stochastic adding machines, Fund. Math., 241(1) (2018), 17–43

work page 2018
[1]

Bayart, E

F. Bayart, E. Matheron, Dynamics of Linear Operators , Cambridge Univ. Press, 2009

work page 2009
[2]

Bezuglyi, O

S. Bezuglyi, O. Karpel, Bratteli diagrams: structure, measures, dynamics. Dynamics and numbers, Contemp. Math. , 669 (2016), 1–36

work page 2016
[3]

X. Buff, A. Ch´ eritat, Quadratic Julia sets with positive area, Ann. Math. , 176 (2012), 673–746

work page 2012
[4]

Caprio, A

D. Caprio, A. Messaoudi, G. Valle, Stochastic adding machines based on Bratteli diagrams, Ann. Inst. Fourier , 70(6) (2020), 2543–2581

work page 2020
[5]

Davies, Linear Operators and Their Spectra , Cambridge University Press, Cam- bridge Studies in Advanced Mathematics 106, 2007

Brian E. Davies, Linear Operators and Their Spectra , Cambridge University Press, Cam- bridge Studies in Advanced Mathematics 106, 2007

work page 2007
[6]

Douady, Disques de Siegel et anneaux de Herman, S´ em

A. Douady, Disques de Siegel et anneaux de Herman, S´ em. Bourbaki 39` eme ann´ ee, 1986/7(677) (1986-87)

work page 1986
[7]

Douady, J

A. Douady, J. Hubbard, Etude dynamique des polynˆ ome complexes , Publication Math´ ematiques d’Orsay, 1984/1985

work page 1984

Show all 22 references

[8]

Fatou, Sur les ´ equations fonctionnelles, Bull

P. Fatou, Sur les ´ equations fonctionnelles, Bull. Soc. Math. Fr. , 47 (1919), 161–271

1919
[9]

Giordano, I

T. Giordano, I. Putnam, C. Skau, Topological orbit equivalence and C∗-crossed products, J. Reine Angew. Math. , 469 (1995), 51–111

1995
[10]

Grosse-Erdmann, A

K. Grosse-Erdmann, A. Peris-Manguillot, Linear Chaos, Springer, 2011

2011
[11]

R. H. Herman, I. Putnam, C. Skau, Ordered Bratteli diagrams, dimension groups, and topological dynamics, Int. J. Math. , 3(6) (1992), 827–864

1992
[12]

Julia, M´ emoire sur l’it´ eration des fonctions rationnelles,J

G. Julia, M´ emoire sur l’it´ eration des fonctions rationnelles,J. Math. Pure Appl. , 8 (1918), 47–245

1918
[13]

Killeen, T

P. Killeen, T. Taylor, A stochastic adding machine and complex dynamics, Nonlinearity, 13 (2000), 1998–1903. 30

2000
[14]

Killeen, T

P. Killeen, T. Taylor, How the Propagation of Error Through Stochastic Counters Affects Time Discrimination and Other Psychophysical Judgments, Psychological Review, 107(3) (2000), 430–459

2000
[15]

A. I. Markushevich, Theory of Functions of Complex Variable, Vol. III , translated by R. A. Silverman, Prentice-Hall Int., London, 1967

1967
[16]

Medynets, Cantor aperiodic systems and Bratteli diagrams, C.R., Math., Acad

K. Medynets, Cantor aperiodic systems and Bratteli diagrams, C.R., Math., Acad. Sci. Paris, 342(1) (2006), 43–46

2006
[17]

Messaoudi, O

A. Messaoudi, O. Sester, G. Valle, Spectrum of stochastic adding machines and fibered Julia sets, Stochastic and Dynamics , 13(3) 2013, 26p

2013
[18]

Messaoudi, D

A. Messaoudi, D. Smania, Eigenvalues of Fibonacci stochastic adding machine, Stochastic and Dynamics , 10(2) (2010), 291–313

2010
[20]

E. M. Stein, R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, 2003

2003
[21]

Vershik, Uniform algebraic approximation of shift and multiplication operators, Dokl

A. Vershik, Uniform algebraic approximation of shift and multiplication operators, Dokl. Acad. Nauk SSSR , 259 (1981), 526–529. (Russian)

1981
[22]

Yosida, Functional Analysis, Springer, 1980

K. Yosida, Functional Analysis, Springer, 1980. 31

1980

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

Messaoudi, G

A. Messaoudi, G. Valle, Spectra of generalized stochastic adding machines, Fund. Math., 241(1) (2018), 17–43

work page 2018

[2] [1]

Bayart, E

F. Bayart, E. Matheron, Dynamics of Linear Operators , Cambridge Univ. Press, 2009

work page 2009

[3] [2]

Bezuglyi, O

S. Bezuglyi, O. Karpel, Bratteli diagrams: structure, measures, dynamics. Dynamics and numbers, Contemp. Math. , 669 (2016), 1–36

work page 2016

[4] [3]

X. Buff, A. Ch´ eritat, Quadratic Julia sets with positive area, Ann. Math. , 176 (2012), 673–746

work page 2012

[5] [4]

Caprio, A

D. Caprio, A. Messaoudi, G. Valle, Stochastic adding machines based on Bratteli diagrams, Ann. Inst. Fourier , 70(6) (2020), 2543–2581

work page 2020

[6] [5]

Davies, Linear Operators and Their Spectra , Cambridge University Press, Cam- bridge Studies in Advanced Mathematics 106, 2007

Brian E. Davies, Linear Operators and Their Spectra , Cambridge University Press, Cam- bridge Studies in Advanced Mathematics 106, 2007

work page 2007

[7] [6]

Douady, Disques de Siegel et anneaux de Herman, S´ em

A. Douady, Disques de Siegel et anneaux de Herman, S´ em. Bourbaki 39` eme ann´ ee, 1986/7(677) (1986-87)

work page 1986

[8] [7]

Douady, J

A. Douady, J. Hubbard, Etude dynamique des polynˆ ome complexes , Publication Math´ ematiques d’Orsay, 1984/1985

work page 1984