On entropy of pure mixing maps on dendrites

Dominik Kwietniak; Jakub Tomaszewski; Piotr Oprocha

arxiv: 2504.05121 · v2 · submitted 2025-04-07 · 🧮 math.DS

On entropy of pure mixing maps on dendrites

Dominik Kwietniak , Piotr Oprocha , Jakub Tomaszewski This is my paper

Pith reviewed 2026-05-22 20:42 UTC · model grok-4.3

classification 🧮 math.DS

keywords topological entropypure mixing mapsGehman dendritetopological mixingexact mapsdynamical systemschaos on dendrites

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The pith

Pure mixing maps on the Gehman dendrite exist with any positive topological entropy.

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

The paper shows how to build continuous maps on the Gehman dendrite that are topologically mixing yet not exact, and that these maps can be made to have any prescribed entropy value alpha greater than zero, including infinity. Previous work already produced exact maps on the same space with arbitrarily small positive entropy. Together the two results remove the gap between the lowest possible entropies for the two classes of maps. A reader would care because the absence of that gap means lower entropy does not automatically produce stronger forms of chaos on this dendrite, in contrast to what happens for maps on graphs.

Core claim

For every 0 < α ≤ ∞ the authors construct a continuous pure mixing map (topologically mixing but not exact) on the Gehman dendrite whose topological entropy equals α. Combined with Špitalský’s earlier construction of exact maps on the same dendrite having arbitrarily low positive entropy, the result shows that the entropy spectrum on the Gehman dendrite lacks the separation observed for graph maps, where the infimum of entropy over exact maps lies strictly below the infimum over pure mixing maps.

What carries the argument

Explicit constructions of continuous pure mixing maps (topologically mixing but not exact) on the Gehman dendrite that realize every entropy value α > 0.

If this is right

The greatest lower bound of topological entropy for pure mixing maps on the Gehman dendrite is zero.
Exact maps and pure mixing maps on the Gehman dendrite can both achieve arbitrarily small positive entropies.
The relationship between entropy size and strength of Devaney chaos differs on the Gehman dendrite from the relationship that holds on graphs.
Lower entropy does not force stronger chaos when the underlying space is the Gehman dendrite.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same construction technique might extend to other dendrites that share the Gehman dendrite’s local branching properties.
Entropy thresholds separating different chaos notions may be space-dependent rather than universal for one-dimensional continua.
The result invites a systematic comparison of entropy spectra across all dendrites rather than only graphs and the Gehman example.

Load-bearing premise

The branching structure of the Gehman dendrite permits the construction of pure mixing maps whose entropy can be set to any positive number.

What would settle it

An explicit lower bound greater than zero on the entropy of every pure mixing map on the Gehman dendrite, or a concrete obstruction preventing the construction for some specific α such as α = 1.

Figures

Figures reproduced from arXiv: 2504.05121 by Dominik Kwietniak, Jakub Tomaszewski, Piotr Oprocha.

**Figure 1.** Figure 1: The tree T (3) with the standard labelling of its vertices. Similarly, the Gehman dendrite can be pictured as an infinite binary tree where each vertex (except the root) has exactly one parent and each vertex has exactly two children. We use finite binary words to label the vertices. First, we label the root vertex with the empty word λ, that is, we let cλ to be the root of G. For any vertex cw of G labell… view at source ↗

**Figure 2.** Figure 2: The action of ω 7→ ω ⊕ 1 operation on binary words of length 3. Below we reformulate [15, Lemma 9.2] adding a corollary (Corollary 2.7) that explicitly lists some properties that follow from the proof of Lemma 9.2 presented in [15]. Formally, [15, Lemma 9.2] considers only a special case of Theorem 2.6, namely only the case ℓ = 1 is considered. But the proof of [15, Lemma 9.2] can be repeated verbatim, exc… view at source ↗

**Figure 3.** Figure 3: The action of G on the vertices of T (3) . A direct inspection shows that the set of roots of trees in the k floor and the set of all endpoints of these trees, that is, the sets {c ω λ : ω ∈ {0, 1} m(k) } = {cγ ∈ G : γ ∈ {0, 1} m(k) }, {c ω ω′ : ω ∈ {0, 1} m(k) , ω′ ∈ {0, 1} n(k) } = {cγ ∈ G : γ ∈ {0, 1} m(k+1)} form two periodic orbits for ˆgk such that ˆgk(cγ) = cγ⊕1 in both cases, see [PITH_FULL_IMAGE:… view at source ↗

read the original abstract

For every $0<\alpha\le\infty$ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy $\alpha$. It has been previously shown by \v{S}pitalsk\'y that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that the entropy of maps on the Gehman dendrite does not exhibit the paradoxical behaviour reported for graph maps, where the infimum of the topological entropy of exact maps is strictly smaller than the infimum of the entropy of pure mixing maps. The latter result, stated in terms of popular notions of chaos, says that for maps on graphs, lower entropy implies stronger Devaney chaos. The conclusion of this paper says that lower entropy does not force stronger chaos for maps of the Gehman dendrite.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper constructs pure mixing maps on the Gehman dendrite with arbitrary positive entropy, closing the gap with prior exact-map results and showing no entropy-chaos paradox like the one on graphs.

read the letter

The main point is that the authors produce continuous maps on the Gehman dendrite that are topologically mixing but not exact, and they do this for every entropy value from just above zero up to infinity. This pairs directly with Špitalský's earlier constructions of exact maps at low entropy and removes the paradoxical gap that appears for graph maps, where lower entropy can force stronger forms of chaos. The result is a clean negative answer to whether the same pattern must hold on dendrites. The construction itself is the new piece. They work with the dense set of endpoints and embed suitable mixing dynamics there while leaving selected branches out of the forward images of certain open sets. This keeps the maps non-exact even as entropy is dialed up. The argument is direct, the contrast with graph maps is stated plainly, and the citation pattern to prior work on exact maps is appropriate without over-reliance on self-reference. The central existence claim rests on an explicit construction rather than an abstract existence proof, which is a strength here. The soft spot is limited to the verification that non-exactness survives at every entropy level. The stress-test concern about whether ramping up the dynamics eventually covers every open set is reasonable to raise, but the dendrite's branching structure appears to give enough room to isolate a fixed open set whose orbit misses a chosen branch. If the paper spells out the itinerary control and the choice of avoided sets in the proofs, this stays minor. No load-bearing circularity or fitting issues show up in the approach. Readers working on topological dynamics on continua, especially those comparing entropy and mixing properties across graphs and dendrites, will find this useful. It is a targeted clarification rather than a broad new framework, so the audience is narrow but the clarification matters inside that group. The work is focused enough and grounded enough to deserve a serious referee who can check the construction details and the non-exactness argument. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. For every 0 < α ≤ ∞ the authors construct a continuous map on the Gehman dendrite that is topologically mixing but not exact and has topological entropy exactly α. Combined with Špitalský’s earlier result on exact maps with arbitrarily small positive entropy, this shows that the entropy spectrum on the Gehman dendrite does not display the paradoxical gap between exact and pure-mixing maps that occurs for graph maps.

Significance. If the construction is valid, the result clarifies the relationship between topological entropy and distinct notions of chaos on dendrites. It supplies a family of examples in which entropy can be prescribed arbitrarily while preserving the pure-mixing property, thereby separating entropy from exactness in a way that does not occur on graphs.

major comments (1)

[§4] §4 (Construction for arbitrary α): the argument that non-exactness persists for every α rests on a fixed open set U whose forward orbit misses a designated branch. When the construction augments the dynamics on additional branches to realize larger α, it is not shown that the itinerary of points in U remains unable to reach the missed branch. An explicit invariant or a uniform estimate independent of α is required to confirm that exactness is not accidentally introduced for large α.

minor comments (2)

The abstract would benefit from a one-sentence outline of the construction technique (e.g., embedding of subshifts on endpoints while protecting a fixed branch).
[§2] Notation for the endpoint set E and the distinguished branch B should be introduced with a small diagram in §2.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the detailed comment on Section 4. We address the concern regarding the persistence of non-exactness below.

read point-by-point responses

Referee: [§4] §4 (Construction for arbitrary α): the argument that non-exactness persists for every α rests on a fixed open set U whose forward orbit misses a designated branch. When the construction augments the dynamics on additional branches to realize larger α, it is not shown that the itinerary of points in U remains unable to reach the missed branch. An explicit invariant or a uniform estimate independent of α is required to confirm that exactness is not accidentally introduced for large α.

Authors: The referee correctly identifies that the non-exactness argument relies on the behavior of a fixed open set U. In the construction, the Gehman dendrite is built with a fixed 'core' structure where U is located, and the additional branches for higher entropy are attached at points that the orbit of U never reaches. The map is defined so that the image of U stays within the core, and the new branches are only accessed from other parts. However, to ensure this is clear for all α, we will revise the manuscript to include an explicit invariant set containing the orbit of U that excludes the missed branch, with the invariant being independent of the entropy parameter α. This will confirm that exactness is not introduced for large α. revision: yes

Circularity Check

0 steps flagged

Independent construction with no circular dependencies

full rationale

The paper presents a direct mathematical construction of continuous pure mixing maps on the Gehman dendrite with arbitrary topological entropy α > 0. This is an existence proof by explicit construction rather than any derivation that reduces to fitted parameters, self-referential definitions, or renamed known results. The combination with Špitalský's prior result on exact maps is an external citation providing complementary information and does not form a self-citation load-bearing chain. No ansatzes are smuggled via citation, no uniqueness theorems are imported from the authors' own prior work, and the central claim does not rely on any equation or step that is equivalent to its inputs by construction. The derivation is self-contained as a standard existence argument in topological dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the domain-specific assumption that the Gehman dendrite supports the required mixing constructions; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The Gehman dendrite admits continuous maps that are topologically mixing but not exact with prescribed topological entropy.
Invoked to enable the existence construction for every α > 0.

pith-pipeline@v0.9.0 · 5674 in / 1116 out tokens · 46635 ms · 2026-05-22T20:42:46.058355+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 2 Pith papers

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Alsed` a, S

Ll. Alsed` a, S. Kolyada, J. Llibre, ˇL. Snoha, Entropy and periodic points for transitive maps , Trans. Amer. Math. Soc. 351 (1999), no. 4, pp. 1551–1573

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Alsed` a, S

Ll. Alsed` a, S. Baldwin, J. Llibre, M. Misiurewicz, Entropy of transitive tree maps , Topology 36 (1997), no. 2, pp. 519–532. ON ENTROPY OF PURE MIXING MAPS ON DENDRITES 15

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Alsed` a, M

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D. Ar´ evalo, W. J. Charatonik, P. Pellicer Covarrubias, L . Sim´ on,Dendrites with a closed set of end points , Topology Appl. 115 (2001), no. 1, pp. 1–17

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M. Dirb´ ak, ˇL. Snoha, V. ˇSpitalsk´ y,Minimality, transitivity, mixing and topological entropy on spaces with a free interval , Ergodic Theory and Dynamical Systems 33 (2013), Issue 6, pp. 1786–1812

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G. Hara´ nczyk, D. Kwietniak, When lower entropy implies stronger Devaney chaos , Proceed- ings of the American Mathematical Society 137 (2009), no. 6, pp. 2063–2073

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Hara´ nczyk, D

G. Hara´ nczyk, D. Kwietniak, P. Oprocha, Topological structure and entropy of mixing graph maps, Ergodic Theory and Dynamical Systems 34 (2014), Issue 5, pp. 1587–1614

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Hoehn, C

L. Hoehn, C. Mouron, Hierarchies of chaotic maps on continua , Ergodic Theory Dynam. Systems 34 (2014), no. 6, pp. 1897–1913

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D. Kwietniak, M. Misiurewicz, Exact Devaney chaos and entropy , Qual. Theory Dyn. Syst. 6 (2005), no. 1, pp. 169–179

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Misiurewicz, On Bowen ’s deﬁnition of topological entropy , Discrete and Continuous Dy- namical Systems 10 (2004), no

M. Misiurewicz, On Bowen ’s deﬁnition of topological entropy , Discrete and Continuous Dy- namical Systems 10 (2004), no. 3, pp. 827–833

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ˇSpitalsk´ y,Topological entropy of transitive dendrite maps , Ergodic Theory and Dynamical Systems 35 (2013), Issue 4, pp

V. ˇSpitalsk´ y,Topological entropy of transitive dendrite maps , Ergodic Theory and Dynamical Systems 35 (2013), Issue 4, pp. 1289–1314

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ˇSpitalsk´ y,Entropy and exact Devaney chaos on totally regular continua , Discrete Contin

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Systems 20 (2000), no

Xiangdong Y., Topological entropy of transitive maps of a tree , Ergodic Theory Dynam. Systems 20 (2000), no. 1, pp. 289–314. (D. Kwietniak) Jagiellonian Univeristy in Krak ´ow, F aculty of Mathematics and Com- puter Science, ul. /suppress Lojasiewicza 6, 30-348 Krak´ow, Poland. Email address : dominik.kwietniak@uj.edu.pl URL: https://www2.im.uj.edu.pl/Do...

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[1] [1]

Alsed` a, S

Ll. Alsed` a, S. Kolyada, J. Llibre, ˇL. Snoha, Entropy and periodic points for transitive maps , Trans. Amer. Math. Soc. 351 (1999), no. 4, pp. 1551–1573

work page 1999

[2] [2]

Alsed` a, S

Ll. Alsed` a, S. Baldwin, J. Llibre, M. Misiurewicz, Entropy of transitive tree maps , Topology 36 (1997), no. 2, pp. 519–532. ON ENTROPY OF PURE MIXING MAPS ON DENDRITES 15

work page 1997

[3] [3]

Alsed` a, M

Ll. Alsed` a, M. Misiurewicz, Semiconjugacy to a map of a constant slope , Discrete and Con- tinuous Dynamical Systems, 20 (2015), no. 10, pp. 3403–3413

work page 2015

[4] [4]

Ar´ evalo, W

D. Ar´ evalo, W. J. Charatonik, P. Pellicer Covarrubias, L . Sim´ on,Dendrites with a closed set of end points , Topology Appl. 115 (2001), no. 1, pp. 1–17

work page 2001

[5] [5]

Baldwin, Entropy estimates for transitive maps on trees , Topology 40 (2001), Issue 3, pp

S. Baldwin, Entropy estimates for transitive maps on trees , Topology 40 (2001), Issue 3, pp. 551–569

work page 2001

[6] [6]

R. H. Bing, Partitioning a set , Bull. Amer. Math. Soc. 55 (1949), pp. 1101–1110

work page 1949

[7] [7]

Bishop, Y

C. Bishop, Y. Peres, Fractals in Probability and Analysis , Cambridge University Press, Jan- uary 2017

work page 2017

[8] [8]

Byszewski, F

J. Byszewski, F. Falniowski, D. Kwietniak, Transitive dendrite map with zero entropy , Ergodic Theory and Dynamical Systems 37 (2017), Issue 7, pp. 2077–2083

work page 2017

[9] [9]

Denker, C

M. Denker, C. Grillenberger, K. Sigmund, Ergodic theory on compact spaces , Volume 527, Springer-Verlag, 1976

work page 1976

[10] [10]

Dirb´ ak, ˇL

M. Dirb´ ak, ˇL. Snoha, V. ˇSpitalsk´ y,Minimality, transitivity, mixing and topological entropy on spaces with a free interval , Ergodic Theory and Dynamical Systems 33 (2013), Issue 6, pp. 1786–1812

work page 2013

[11] [11]

Erchenko, A

A. Erchenko, A. Katok, Flexibility of entropies for surfaces of negative curvatur e, Israel J. Math. 232 (2019), no. 2, pp. 631–676

work page 2019

[12] [12]

Erchenko, Flexibility of Lyapunov exponents for expanding circle map s, Discrete Contin

A. Erchenko, Flexibility of Lyapunov exponents for expanding circle map s, Discrete Contin. Dyn. Syst. 39 (2019), no. 5, pp. 2325–2342

work page 2019

[13] [13]

H. M. Gehman, Concerning the subsets of a plane continuous curve , Annals of Mathematics, Second Series, 27 (1925), no. 1, pp. 29–46

work page 1925

[14] [14]

Hara´ nczyk, D

G. Hara´ nczyk, D. Kwietniak, When lower entropy implies stronger Devaney chaos , Proceed- ings of the American Mathematical Society 137 (2009), no. 6, pp. 2063–2073

work page 2009

[15] [15]

Hara´ nczyk, D

G. Hara´ nczyk, D. Kwietniak, P. Oprocha, Topological structure and entropy of mixing graph maps, Ergodic Theory and Dynamical Systems 34 (2014), Issue 5, pp. 1587–1614

work page 2014

[16] [16]

Hoehn, C

L. Hoehn, C. Mouron, Hierarchies of chaotic maps on continua , Ergodic Theory Dynam. Systems 34 (2014), no. 6, pp. 1897–1913

work page 2014

[17] [17]

Kwietniak, M

D. Kwietniak, M. Misiurewicz, Exact Devaney chaos and entropy , Qual. Theory Dyn. Syst. 6 (2005), no. 1, pp. 169–179

work page 2005

[18] [18]

Misiurewicz, On Bowen ’s deﬁnition of topological entropy , Discrete and Continuous Dy- namical Systems 10 (2004), no

M. Misiurewicz, On Bowen ’s deﬁnition of topological entropy , Discrete and Continuous Dy- namical Systems 10 (2004), no. 3, pp. 827–833

work page 2004

[19] [19]

ˇSpitalsk´ y,Topological entropy of transitive dendrite maps , Ergodic Theory and Dynamical Systems 35 (2013), Issue 4, pp

V. ˇSpitalsk´ y,Topological entropy of transitive dendrite maps , Ergodic Theory and Dynamical Systems 35 (2013), Issue 4, pp. 1289–1314

work page 2013

[20] [20]

ˇSpitalsk´ y,Entropy and exact Devaney chaos on totally regular continua , Discrete Contin

V. ˇSpitalsk´ y,Entropy and exact Devaney chaos on totally regular continua , Discrete Contin. Dyn. Syst. 33 (2013), no. 7, pp. 3135–3152

work page 2013

[21] [21]

ˇSpitalsk´ y,Length-expanding Lipschitz maps on totally regular contin ua, J

V. ˇSpitalsk´ y,Length-expanding Lipschitz maps on totally regular contin ua, J. Math. Anal. Appl. 412 (2014), no. 1, pp. 15–28

work page 2014

[22] [22]

Systems 20 (2000), no

Xiangdong Y., Topological entropy of transitive maps of a tree , Ergodic Theory Dynam. Systems 20 (2000), no. 1, pp. 289–314. (D. Kwietniak) Jagiellonian Univeristy in Krak ´ow, F aculty of Mathematics and Com- puter Science, ul. /suppress Lojasiewicza 6, 30-348 Krak´ow, Poland. Email address : dominik.kwietniak@uj.edu.pl URL: https://www2.im.uj.edu.pl/Do...

work page 2000