Thermodynamic formalism for coarse expanding dynamical systems

Anna Zdunik; Feliks Przytycki; Giulio Tiozzo; Mariusz Urbanski; Tushar Das

REVIEW 2 major objections 5 minor 33 references

Thermodynamic formalism for coarse expanding dynamical systems

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Weakly coarse expanding dynamical systems admit unique equilibrium states for every Hölder potential, along with the central limit theorem, law of iterated logarithm, exponential decay of correlations, and large deviations.

desk verdict Genuinely new and mostly sound paper on equilibrium states for weakly coarse expanding systems, but Theorem 1.1(5) as printed is an ill-posed large-deviation statement that needs to be corrected. read the letter →

arxiv 1908.08270 v3 pith:QN3EEAAP submitted 2019-08-22 math.DS

Tushar Das , Feliks Przytycki , Giulio Tiozzo , Mariusz Urbanski , Anna Zdunik This is my paper · ORCID

classification math.DS MSC 37D3537F1037A5037B10

keywords thermodynamicformalismequilibriumstatescoarseexpandingdynamicalsystemspostcriticallyinfinitecentrallimittheoremlawofiteratedlogarithmlargedeviationsgeometriccodingtree

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

The paper brings thermodynamic formalism to a broad class of expanding, possibly postcritically infinite maps: it proves that every Hölder continuous potential on the repellor has a unique equilibrium state. The class of weakly coarse expanding systems includes previously studied expanding sphere maps and coarse expanding conformal systems, but drops finiteness of the postcritical set and does not assume conformality, holomorphy, or smoothness. The proof encodes the dynamics by a geometric coding tree that semiconjugates the system to a full shift, transfers equilibrium states and statistical laws from the shift, and then handles periodic branch points by blowing them up to circles. If the theorems are correct, any Hölder observable on such a repellor is statistically tame: Gaussian fluctuations, iterated-logarithm growth, exponential decay of correlations, and large deviations all follow automatically.

What carries the argument

The argument runs through two constructions. First, paths in $W_0$ joining a basepoint to each of its $d$ preimages and avoiding the countable post-branch set build a geometric coding tree: the branches give a Hölder semiconjugacy $\pi$ from the full shift $\Sigma$ onto the repellor $X$, and a no-entropy-drop lemma (the number of depth-$n$ cylinders over a point grows subexponentially when no critical point is periodic) lets equilibrium states and statistical laws be pulled back from the shift. Second, for periodic critical points on the sphere, each such point and its grand orbit is blown up to a circle, giving a system without periodic critical points on a carpet-like space; a metrization lemma produces the exponentially contracting metric on the new repellor.

What would settle it

Find (or prove impossible) a weakly coarse expanding system on a compact, locally connected space that is path connected but not strongly path connected—for instance, a finite tree-like branched cover—and test whether a Hölder potential on its repellor has a unique equilibrium state. If such a system admits two equilibrium states, strong path connectivity is a necessary assumption, not an artifact of the proof.

Watch

Extended reading notes

Core claim

The paper introduces weakly coarse expanding dynamical systems—continuous finite branched coverings of locally connected, strongly path connected spaces that expand in a weak metric sense—and proves that whenever no critical point is periodic, every Hölder continuous potential $\phi$ on the repellor $X$ has exactly one equilibrium state, and every Hölder observable $\psi$ satisfies the central limit theorem, the law of the iterated logarithm, exponential decay of correlations, and the large deviation principle. If $W_0$ is an open subset of the 2-sphere, the same conclusions hold even with periodic (repelling) branch points: the authors blow up each periodic critical point and its grand orbit to circles, obtaining a carpet-like space with no periodic critical points, then run the non-critical argument.

Load-bearing premise

The entire construction rests on the phase space being strongly path connected: after deleting any countable set of points, the space must remain path connected, because the coding tree is built from paths that avoid the countable post-branch set. A merely path connected space, such as a tree, can be disconnected by removing a single point.

Editorial extensions

If this is right

Every Hölder potential on the repellor has a unique equilibrium state, and it is the pushforward of the unique Gibbs state for a Hölder potential on the full shift.
Every Hölder observable obeys the central limit theorem, law of iterated logarithm, exponential decay of correlations, and a large deviation principle with rate function given by topological pressure.
The variance in the CLT is zero exactly when the centered observable is a continuous coboundary, and two potentials share an equilibrium state exactly when they differ by a coboundary plus a constant.
All of this remains true on an open subset of the 2-sphere even when periodic branch points are present.
The results apply to a class of examples—iterated function systems, semigroup skew products, maps on carpet-like fractals, expanding polynomial-like maps—that need not be conformal, holomorphic, or postcritically finite.

Reading between the lines

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

This suggests that strong path connectivity could be relaxed to the existence of a finite covering by path connected sets whose intersections remain path connected after removing countable sets; if a coding tree could be replaced by such a covering, the theorems would extend to dendrites and trees.
The paper's criterion for zero variance in the CLT is immediately testable numerically on a postcritically infinite rational map: compute sample variances of a Hölder observable along orbits and check whether they vanish precisely for coboundaries.
The blow-up construction points to a two-way dictionary between periodic branch points and invariant circles, which could generate new examples by starting from a circle map on a carpet and collapsing the invariant circles.
Since the proofs use only oscillation decay along pullback covers, the results should extend to observables with weaker moduli of continuity than Hölder, as long as the topologically Hölder condition is met.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Plainly: this is a substantial paper and the main results are mostly believable. The new class of weakly coarse expanding systems is a useful generalization, and the geometric coding tree plus the blow-up of periodic critical points is a genuine technical achievement. The no-entropy-drop lemma (Lemma 3.2) is clean, and the two independent constructions of an exponentially contracting metric (Frink's lemma and the explicit metric in Appendix A) show real care. I found no circularity: the shift-space facts cited to [24] and Bowen are standard and independently verifiable, and the self-citations are appropriate.

The trouble is Theorem 1.1(5), the large-deviation claim. As printed, the event contains a free 'tψ' on the right side and the displayed rate also contains 'tψ', so the claimed limit depends on x rather than being a number. That is ill-posed. Worse, even a naive guess that 'tψ' should be 't' does not fix the mathematics: the expression -t∫ψ dµ + P(φ+tψ) - P(φ) is not the level-1 rate function for the empirical average under an equilibrium state; for a full shift with φ≡0 and ψ the first-coordinate indicator, the displayed value has the wrong sign and magnitude. The proof in Section 4 only says the sequence satisfies the LDP by transfer from the shift; it never derives the displayed rate. So the theorem needs a genuine correction or replacement, not a one-character typo fix. I suspect this is repairable, but it is a real defect in a headline claim.

The rest of the paper holds up well. The blow-up construction in Section 5 is intricate and I did not independently verify every topological claim, but the overall architecture is coherent and the exposition is honest about hypotheses. The strong path connectedness assumption (Definition 2.3) is restrictive and rules out simple examples like trees, but it is explicitly flagged and is a reasonable price for the coding tree.

This paper deserves a serious referee. I would send it to peer review rather than desk reject, but the referee should insist that Theorem 1.1(5) be fixed or removed, and that the proof provide the actual rate function. After that, I would expect the paper to be accepted.

Referee Report

2 major / 5 minor

Summary. The paper develops thermodynamic formalism for weakly coarse expanding systems: continuous finite branched covers of locally compact strongly path connected spaces satisfying topological expansion and irreducibility. Its main results assert existence and uniqueness of equilibrium states for Hölder potentials and the validity of the central limit theorem, law of the iterated logarithm, exponential decay of correlations, and a large deviation principle, both for systems without periodic critical points (Theorem 1.1) and for open subsets of S² with periodic critical points allowed (Theorem 1.2). The strategy is to encode the system by a geometric coding tree semiconjugate to the full shift, prove that entropy does not drop under the coding map, and transfer statistical laws from the shift; in the S² case, periodic critical points and their grand orbits are blown up to circles, producing a weakly coarse expanding system without periodic critical points.

Significance. The framework unifies and extends earlier work on expanding Thurston maps and coarse expanding conformal systems, allowing postcritically infinite behavior and non-conformal dynamics. The paper contains detailed constructive proofs, including two independent constructions of an exponentially contracting metric (via Frink's lemma and an explicit chain metric), and the coding-tree/no-entropy-drop argument is robust. If the statistical laws are stated correctly, the paper would be a valuable contribution to the thermodynamic formalism of non-uniformly hyperbolic systems. The present version, however, contains a defective large-deviation statement in the headline theorem, so the paper needs revision.

major comments (2)

[Theorem 1.1(5); Section 4, proof of Theorem 1.1(1)-(5)] The displayed large-deviation assertion in Theorem 1.1(5) is not a well-formed statement. The event contains the expression tψ(x) on the right-hand side of the inequality, and the right-hand side of the limit also contains tψ, so the claimed limit is point-dependent and the equality is not meaningful as a statement about a fixed real t. Moreover, even under the minimal correction tψ→0 on the right-hand side, the formula does not give the rate inherited from the full shift. For the full shift on two symbols, φ≡0, ψ=1_{first coordinate}, and t=0.1, the exact rate for {S_nψ ≥ 0.6n} is −D(0.6‖0.5) ≈ −0.0202, whereas the displayed right-hand side evaluates to P(0.1ψ)−P(0)−0.1∫ψ ≈ +0.0012 under the first correction and to ≈ +0.1012 if tψ is replaced by t. Neither equals the Cramér rate, and the sign is wrong in the first interpretation. The transfer argument in Section 4 can at best yield the standard LDP for the full shift, so Theorem 1.1(5) should be replaced by a correct statement, for example with the rate given by the Legendre transform associated to t↦P(φ+tψ)−P(φ)−t∫ψ. This is a load-bearing defect in the main theorem, not a typographical issue in a proof.
[Section 5.1, proof of Theorem 1.2(1)-(5)] Theorem 1.2 asserts that claim (5) of Theorem 1.1 holds also in the presence of periodic critical points, and the proof in Section 5.1 says that statistical laws follow from the shift by pushforward. Since claim (5) as printed is ill-posed and the displayed rate is not the shift rate, this proof cannot establish the stated theorem. The correction to Theorem 1.1(5) must be propagated to Theorem 1.2; otherwise the main theorems remain defective even if the rest of the thermodynamic formalism argument is sound.

minor comments (5)

[Section 4, Lemma 4.3] In the proof of Lemma 4.3, the equality "π⋆~µ = ~µ" should read "π⋆~µ = µ"; the current text appears to be a typo.
[Section 5, after Proposition 5.1] The symbol ~W0 is used both for the blown-up space and for the lifted cover ("Denote this cover by ~W0"), which creates ambiguity in Lemma 5.3; a different letter, such as ~U0, should be used for the cover.
[Theorem 1.1(5)] The function sgn(t) is undefined at t=0; the corrected large-deviation statement should specify t∈R\{0} or include a separate convention for t=0.
[Section 2.6, proof of Proposition 2.16] The phrase "a curve which is the branch of f^{-(n-1)}(γ_{i_n})" is not formally precise, since f^{-(n-1)} of a curve is not a single curve; it should say that γ_n(α) is the lift of γ_{i_n} by f^{n-1} starting at z_{n-1}(α).
[Appendix A, after equation (14)] The parameter c is introduced as "a constant which will be determined later" but the proof only chooses it in Step 3; the text should state explicitly that c can be made small enough to satisfy both cd²e^{α₂}<1 and cd²<1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the coding-map reduction to the full shift is self-contained; self-citations are to established, independently verified thermodynamic formalism.

full rationale

The paper's derivation is not circular. The central reduction constructs an explicit Holder semiconjugacy pi : Sigma -> X (Proposition 2.16), proves that entropy does not drop under pi (Lemma 3.3), compares pressures (Lemma 4.2), and transfers equilibrium states and statistical laws from the full shift. These steps are proven in the paper, not assumed. The shift-space CLT, LIL, exponential decay, and large deviations are cited to [24] (Przytycki-Urbanski) and [10]; although [24] is by two of the present authors, these are published, standalone classical results with independent proofs, so the citation is real evidence and does not make the argument circular. No parameter is fitted to data, and no prediction is equivalent to an input by construction. The large-deviation display in Theorem 1.1(5) is ill-posed as printed (the right-hand side depends on the point x through the term t psi(x)), but that is a correctness/typographical defect, not a circularity: the proof does not derive that displayed formula from itself.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

The paper's central theorems rest on a standard body of ergodic theory and topology: the variational principle, the existence and uniqueness of equilibrium states and statistical laws for Holder potentials on the full shift, Frink's metrization lemma, and topological facts about the plane and 2-sphere. The specifically new domain assumptions are the axioms of a weakly coarse expanding system (Expansion, Irreducibility/leo, finite branch set, repellor not a point) and strong path connectedness of the ambient space. The blow-up construction introduces an auxiliary Sierpinski-carpet space but no unexplained physical or dynamical postulate.

assumptions (8)

standard math Variational principle for topological pressure on compact metric spaces.
Used in Section 2.5 and in Lemma 4.2 / Proposition 5.5 to convert equality of pressure into existence of equilibrium states.
standard math Thermodynamic formalism for the one-sided full shift on d symbols: unique equilibrium states, Gibbs property, CLT, LIL, exponential decay of correlations, large deviations for Holder potentials.
Invoked in the proof of Theorem 1.1 (1)-(5) and Theorem 1.2, with references to Bowen [5], Przytycki-Urbanski [24], and Denker-Kessebohmer [10].
standard math Riesz extension theorem and Riesz representation theorem.
Used in Lemma 4.1 to prove that every invariant measure on X has a sigma-invariant lift to the shift space.
standard math Frink's metrization lemma for constructing metrics from expanding sequences of covers.
Used in Theorem 2.12 to prove existence of exponentially contracting metrics and in Lemma 5.3 for the blown-up space.
standard math Topological facts: Jordan curve theorem, Caratheodory's theorem on extensions of Riemann maps, Whyburn's theorem on open maps on 2-manifolds.
Used in Lemma 5.2 to derive the local model at a fixed critical point and in the blow-up construction (Section 5 and Appendix B).
domain assumption The space W0 is strongly path connected (Definition 2.3), i.e. removing any countable set leaves it path connected.
Needed in Proposition 2.16 to choose curves gamma_i from a basepoint to each preimage avoiding the countable post-branch set P_f; the coding tree cannot be built without it.
domain assumption The system satisfies the weakly coarse expanding axioms: [Expansion], [Irreducibility] (leo), finite branch set, and repellor X not a single point (Definition 2.10).
These axioms define the class studied in the paper; the theorems are stated for exactly this class.
domain assumption For Theorem 1.2, W0 is an open subset of the 2-sphere S^2 with Euclidean topology.
The blow-up construction producing a Sierpinski carpet and the finiteness of the branch set require the ambient 2-sphere.

invented entities (1)

Blown-up space ~W0 (and repellor Y), in which each point in the grand orbit of a periodic critical point is replaced by a circle, producing a Sierpinski carpet.
purpose: Converts a weakly coarse expanding system with periodic critical points into one without periodic critical points, so the coding-tree and shift-space thermodynamic formalism can be applied.
This is an auxiliary mathematical construction used in the proof of Proposition 5.1 and Theorem 1.2. It has no observable consequences outside the proof and is not an empirical postulate.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Thermodynamic formalism for coarse expanding dynamical systems." pith.science (2026). https://pith.science/paper/QN3EEAAP

@misc{pith2026190808270,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic formalism for coarse expanding dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QN3EEAAP}},
  note         = {Machine review of arXiv:1908.08270}
}

read the original abstract

We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all such results even in presence of periodic (repelling) branch points.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[24]

Przytycki, M

F. Przytycki, M. Urba´ nski, Conformal Fractals: Ergodic Theory Methods , London Mathematical Society Lecture Note Series (371), Cambridge University Press, 2010

work page 2010
[1]

Atnip, H

J. Atnip, H. Sumi, M. Urba´ nski, The Dynamics and Geometry of Semi–Hyperbolic Rational Semi groups, in preparation

work page
[2]

Blokh, C

A. Blokh, C. Cleveland, M. Misiurewicz, Expanding polymodials, in Modern dynamical systems and applications , 253–270, Cambridge Univ. Press, Cambridge, 2004

work page 2004
[3]

Blokh, C

A. Blokh, C. Cleveland, M. Misiurewicz, Julia sets of expanding polymodials , Ergodic Theory Dynam. Systems 25 (2005), 1691–1718

work page 2005
[4]

M. Bonk, D. Meyer, Expanding Thurston maps , Mathematical Surveys and Monographs 225, American Mathe- matical Society, 2017

work page 2017
[5]

Bowen, Equilibrium States and the Ergodic Theory of Anosov Diﬀeomor phisms, Lecture Notes in Mathematics 470, Springer, 1975

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diﬀeomor phisms, Lecture Notes in Mathematics 470, Springer, 1975

work page 1975
[6]

J. W. Cannon, W. J. Floyd, W. R. Parry, Finite subdivision rules , Conform. Geom. Dyn. 5 (2001), 153–196

work page 2001
[7]

Collet, J.-P

P. Collet, J.-P. Eckmann, Positive Lyapunov exponents and absolute continuity for ma ps of the interval , Ergodic Theory Dynam. Systems 3 (1983), 13–46

work page 1983

Show all 33 references

[8]

Comman, J

H. Comman, J. Rivera-Letelier, Large deviation principles for non-uniformly hyperbolic r ational maps , Ergodic Theory Dynam. Systems 31 (2011), no. 2, 321–349

2011
[9]

Constantin, B

A. Constantin, B. Kolev, The theorem of Ker´ ekj´ art´ o on periodic homeomorphisms ofthe disc and the sphere , Enseign. Math. (2) 40 (1994), no. 3-4, 193–204

1994
[10]

Denker, M

M. Denker, M. Kesseb¨ ohmer,Thermodynamic formalism, large deviation, and multifract als, in Stochastic Climate Models, Progr. Probab. 49 (2001), 159–170

2001
[11]

Engelking, General Topology, Heldermann, Berlin, 1989

R. Engelking, General Topology, Heldermann, Berlin, 1989

1989
[12]

Ero˘ glu, S

K. Ero˘ glu, S. Rohde, B. Solomyak, Quasisymmetric conjugacy between quadratic dynamics and i terated function systems, Ergodic Theory Dynam. Systems, 30 (2010), 1665–1684

2010
[13]

A. H. Frink, Distance functions and metrization problem , Bull. Amer. Math. Soc. 43 (1937), 133–142

1937
[14]

Y. Gao, P. Ha ¨ ıssinsky, D. Meyer, J. Zeng, Invariant Jordan curves of Sierpi´ nski carpet rational map s, Ergodic Theory Dynam. Systems 38 (2018), no. 2, 583–600

2018
[15]

Ha ¨ ıssinsky, K

P. Ha ¨ ıssinsky, K. M. Pilgrim,Coarse expanding conformal dynamics , Ast´ erisque 325 (2009), viii+139 pp

2009
[16]

Ha ¨ ıssinsky, K

P. Ha ¨ ıssinsky, K. M. Pilgrim, Examples of coarse expanding conformal maps , Discrete Contin. Dyn. Syst. 32 (2012), no. 7, 2403–2416

2012
[17]

Inoquio-Renteria, J

I. Inoquio-Renteria, J. Rivera-Letelier, A characterization of hyperbolic potentials of rational ma ps, Bull. Braz. Math. Soc. (N.S.) 43 (2012), no. 1, 99–127

2012
[18]

Li, Weak expansion properties and large deviation principles f or expanding Thurston Maps , Adv

Z. Li, Weak expansion properties and large deviation principles f or expanding Thurston Maps , Adv. Math. 285 (2015), 515–567

2015
[19]

Li, Periodic points and the measure of maximal entropy of an expa nding Thurston map , Trans

Z. Li, Periodic points and the measure of maximal entropy of an expa nding Thurston map , Trans. Amer. Math. Soc. 368 (2016), 8955–8999

2016
[20]

Li, Equilibrium states for expanding Thurston maps , Comm

Z. Li, Equilibrium states for expanding Thurston maps , Comm. Math. Phys. 357 (2018), 811–872

2018
[21]

Li, Ergodic theory of expanding Thurston maps , The Atlantis Series in Dynamical Systems, volume 4, Atlant is Press (Springer), 2017

Z. Li, Ergodic theory of expanding Thurston maps , The Atlantis Series in Dynamical Systems, volume 4, Atlant is Press (Springer), 2017

2017
[22]

Przytycki, Hausdorﬀ dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent

F. Przytycki, Hausdorﬀ dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), 161–179

1985
[23]

Przytycki, Thermodynamic formalism methods in one-dimensional real a nd complex dynamics , Proceedings of the International Congress of Mathematicians 2018, Rio d e Janeiro, vol

F. Przytycki, Thermodynamic formalism methods in one-dimensional real a nd complex dynamics , Proceedings of the International Congress of Mathematicians 2018, Rio d e Janeiro, vol. 2, 2081–2106

2018
[25]

Przytycki, M

F. Przytycki, M. Urba´ nski, A. Zdunik. Harmonic, Gibbs and Hausdorﬀ Measures on Repellers for Holo morphic Maps, I , Ann. of Math. (2) 130, no. 1 (1989), 1–40

1989
[26]

Ruelle, Thermodynamic formalism , Vol

D. Ruelle, Thermodynamic formalism , Vol. 5. Encyclopedia of Mathematics and its Applications, Addison- Wesley, Reading, Mass., 1978

1978
[27]

Simon, Real Analysis: A comprehensive course in Analysis, Part I , American Math

B. Simon, Real Analysis: A comprehensive course in Analysis, Part I , American Math. Society, Providence, RI, 2015

2015
[28]

Ja. G. Sinai, Gibbs measures in ergodic theory , Uspekhi Mat. Nauk. 27.4 (166) (1972), 21–64

1972
[29]

E. H. Spanier, Algebraic Topology, McGraw Hill Inc., 1966

1966
[30]

Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems , Invent

M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems , Invent. Math. 111 (1993), 113–137

1993
[31]

Walters, Relative Pressure, Relative Equilibrium States, Compensa tion Functions and Many-to-One Codes Between Subshifts , Trans

P. Walters, Relative Pressure, Relative Equilibrium States, Compensa tion Functions and Many-to-One Codes Between Subshifts , Trans. Amer. Math. Soc. 296 (1986), no. 1, 1–31

1986
[32]

G. T. Whyburn, Open Mappings on 2-Dimensional Manifolds , Journal of Mathematics and Mechanics 10 (1961), no. 1, 181–197. 30 T. DAS, F. PRZYTYCKI, G. TIOZZO, M. URBA ´NSKI, AND A. ZDUNIK

1961
[33]

G. T. Whyburn, Analytic topology, American Mathematical Society, New York, 1942. Department of Mathematics and Statistics, University of Wi sconsin - La Crosse, 1725 State Street, La Crosse, WI 54601, USA Email address : tdas@uwlax.edu Institute of Mathematics, Polish Academy ...

1942

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [24]

Przytycki, M

F. Przytycki, M. Urba´ nski, Conformal Fractals: Ergodic Theory Methods , London Mathematical Society Lecture Note Series (371), Cambridge University Press, 2010

work page 2010

[2] [1]

Atnip, H

J. Atnip, H. Sumi, M. Urba´ nski, The Dynamics and Geometry of Semi–Hyperbolic Rational Semi groups, in preparation

work page

[3] [2]

Blokh, C

A. Blokh, C. Cleveland, M. Misiurewicz, Expanding polymodials, in Modern dynamical systems and applications , 253–270, Cambridge Univ. Press, Cambridge, 2004

work page 2004

[4] [3]

Blokh, C

A. Blokh, C. Cleveland, M. Misiurewicz, Julia sets of expanding polymodials , Ergodic Theory Dynam. Systems 25 (2005), 1691–1718

work page 2005

[5] [4]

M. Bonk, D. Meyer, Expanding Thurston maps , Mathematical Surveys and Monographs 225, American Mathe- matical Society, 2017

work page 2017

[6] [5]

Bowen, Equilibrium States and the Ergodic Theory of Anosov Diﬀeomor phisms, Lecture Notes in Mathematics 470, Springer, 1975

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diﬀeomor phisms, Lecture Notes in Mathematics 470, Springer, 1975

work page 1975

[7] [6]

J. W. Cannon, W. J. Floyd, W. R. Parry, Finite subdivision rules , Conform. Geom. Dyn. 5 (2001), 153–196

work page 2001

[8] [7]

Collet, J.-P

P. Collet, J.-P. Eckmann, Positive Lyapunov exponents and absolute continuity for ma ps of the interval , Ergodic Theory Dynam. Systems 3 (1983), 13–46

work page 1983