pith. sign in

arxiv: 2605.18278 · v1 · pith:I5W36ZKMnew · submitted 2026-05-18 · 🧮 math.DS

Isomoprhism of generalized Bratteli diagrams

Olena Karpel This is my paper

Pith reviewed 2026-05-20 00:04 UTC · model grok-4.3

classification 🧮 math.DS
keywords generalized Bratteli diagramsisomorphismirreducible diagramspath spacetail equivalence relationtopological dynamics
0
0 comments X

The pith

Every generalized Bratteli diagram is isomorphic to an irreducible generalized Bratteli diagram.

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

The paper defines isomorphism for generalized Bratteli diagrams and studies which features of the diagram and its path space survive the equivalence. It proves that any diagram can be replaced by an isomorphic irreducible version. The argument preserves the tail equivalence relation on the infinite paths through the diagram. The authors then single out completely irreducible diagrams, those that admit no further reduction, and tie this property to topological features of the path space.

Core claim

We show that every generalized Bratteli diagram is isomorphic to an irreducible generalized Bratteli diagram. We introduce the notion of a completely irreducible generalized Bratteli diagram, namely, a diagram that is isomorphic only to irreducible generalized Bratteli diagrams. We establish connections between this notion and the topological properties of the path space of a generalized Bratteli diagram and its tail equivalence relation.

What carries the argument

Isomorphism of generalized Bratteli diagrams that preserves the tail equivalence relation on the path space.

If this is right

  • Any dynamical or topological property of the path space that is preserved by isomorphism can be studied using only irreducible diagrams.
  • The tail equivalence relation remains unchanged under the reduction to an irreducible form.
  • Completely irreducible diagrams are those whose only isomorphic copies are already irreducible.
  • Several concrete classes of generalized Bratteli diagrams are shown to illustrate the reduction and the topological links.

Where Pith is reading between the lines

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

  • Studies of orbit equivalence or symbolic dynamics built on these diagrams can assume the diagram is irreducible without loss of generality.
  • Complete irreducibility may serve as a minimality condition that distinguishes diagrams with unique representation up to isomorphism.
  • The link between irreducibility and path-space topology suggests a route to classifying diagrams by their associated equivalence relations.

Load-bearing premise

The chosen definitions of generalized Bratteli diagrams, irreducibility, and isomorphism always permit reduction to an irreducible diagram without changing the tail equivalence relation.

What would settle it

An explicit generalized Bratteli diagram together with a proof that no isomorphic copy of it is irreducible, or that every isomorphic copy alters the tail equivalence relation on the path space.

Figures

Figures reproduced from arXiv: 2605.18278 by Olena Karpel.

Figure 1
Figure 1. Figure 1: Isomorphic generalized Bratteli diagrams B and B′ . a (n) ij are elements of the matrix A n . A stationary generalized Bratteli diagram is irreducible if and only if the corresponding incidence matrix is irreducible. Denote p(i) = gcd{n ∶ a (n) ii > 0}. Then p(i) is called the period of index i. For an irreducible matrix A, the periods of all indices are the same and called the period of A. An irreducible … view at source ↗
Figure 2
Figure 2. Figure 2: One-sided stationary generalized Bratteli diagram BRS which corresponds to the renewal shift. In other words, the vertex 1 of each level is connected by a single edge to all vertices on the level below, and for n > 1, the vertex n is connected by a singe edge to the vertex (n−1) on the level below (see [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A stationary one-sided BIO that does not have finite invariant measures. The structure of the two-sided diagram B̃IO is more complicated than the one of BIO since not every path in XB̃IO is eventually vertical. Since F̃n has only values 1 right above the main diagonal, the set of slanted paths is countable (see also [BJKS25, Proposition 5.5]). Proposition 3.12. Both one-sided and two-sided Bratteli diagram… view at source ↗
Figure 4
Figure 4. Figure 4: One-sided stationary generalized Bratteil diagram B∞. The diagram B∞ was introduced in [BKKW26, Section 8] as an example of a diagram which has uncountably many probability ergodic invariant measures, and one can explicitly [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A stationary diagram with non-transitive tail-equivalence relation. We generalize the Proposition 4.3 to obtain the following theorem: Theorem 4.5. Every generalized Bratteli diagram is isomorphic to an irreducible generalized Bratteli diagram. Proof. Let B be a generalized Bratteli diagram. Then XB is a Polish space, hence, there exist a countable set of paths {x i } ∞ i=0 ⊂ XB which is dense in XB. We sh… view at source ↗
read the original abstract

We study the notion of isomorphism for generalized Bratteli diagrams and investigate properties preserved under isomorphism. We show that every generalized Bratteli diagram is isomorphic to an irreducible generalized Bratteli diagram. We introduce the notion of a completely irreducible generalized Bratteli diagram, namely, a diagram that is isomorphic only to irreducible generalized Bratteli diagrams. We establish connections between this notion and the topological properties of the path space of a generalized Bratteli diagram and its tail equivalence relation. We examine in detail several classes of generalized Bratteli diagrams that illustrate these results.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript studies isomorphisms of generalized Bratteli diagrams, proving that every such diagram is isomorphic to an irreducible generalized Bratteli diagram. It introduces the notion of a completely irreducible generalized Bratteli diagram (one isomorphic only to irreducible diagrams) and relates this to topological properties of the path space and its tail equivalence relation. Several concrete classes of generalized Bratteli diagrams are examined to illustrate the results.

Significance. If the central results hold, the explicit reduction to an irreducible form supplies a canonical representative that preserves the tail equivalence relation on the path space, which could streamline classification and study of the associated dynamical systems. The introduction of completely irreducible diagrams and the examination of concrete classes provide concrete examples that strengthen the utility of the framework.

major comments (1)
  1. §3, Theorem on reduction to irreducible diagram: the construction removes selected vertices/edges to obtain an irreducible diagram while inducing a homeomorphism on path spaces; a short verification that this map remains bijective when the original diagram has vertices of unbounded degree would strengthen the argument, as the current sketch relies on the tail-equivalence preservation without an explicit inverse map in that case.
minor comments (3)
  1. Introduction: a one-sentence outline of the proof strategy for the main isomorphism theorem would help readers navigate the subsequent sections.
  2. Notation: the symbols for the path space and tail equivalence relation are introduced gradually; collecting them in a single preliminary subsection would improve readability.
  3. Examples section: the diagrams illustrating completely irreducible cases could include a small table summarizing which topological properties (e.g., minimality of the equivalence relation) are verified for each class.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive suggestion. We address the single major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

  1. Referee: §3, Theorem on reduction to irreducible diagram: the construction removes selected vertices/edges to obtain an irreducible diagram while inducing a homeomorphism on path spaces; a short verification that this map remains bijective when the original diagram has vertices of unbounded degree would strengthen the argument, as the current sketch relies on the tail-equivalence preservation without an explicit inverse map in that case.

    Authors: We appreciate the referee's observation. The construction in Section 3 defines the reduced diagram by excising vertices and edges that violate irreducibility while ensuring that the induced map on path spaces preserves tail equivalence classes. Bijectivity holds because every infinite path in the original diagram that eventually follows a surviving edge sequence corresponds uniquely to a path in the reduced diagram, and the converse inclusion follows from the fact that removed vertices have no infinite paths remaining after a finite stage. Nevertheless, we agree that an explicit verification of bijectivity for the unbounded-degree case would make the argument more transparent and self-contained. In the revised version we will add a short paragraph constructing the inverse map directly: given a path in the reduced diagram, extend it by the unique tail-equivalent continuation in the original diagram using the definition of the path space topology. This addition strengthens the exposition without changing the theorem statement or its proof strategy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes that every generalized Bratteli diagram is isomorphic to an irreducible one via an explicit construction that induces a homeomorphism on path spaces preserving the tail equivalence relation. This rests on standard definitions of generalized Bratteli diagrams, isomorphism, and irreducibility drawn from prior literature in symbolic dynamics and Cantor systems, without any reduction of the central claim to fitted inputs, self-definitional loops, or load-bearing self-citations. The derivation is self-contained as a direct combinatorial/topological argument that does not presuppose the result it proves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on standard background from dynamical systems theory and introduces new definitions without new physical postulates or fitted parameters.

axioms (1)
  • standard math Standard properties of path spaces, tail equivalence relations, and Bratteli diagrams from prior literature in topological dynamics
    Invoked implicitly when discussing preserved properties under isomorphism and connections to the path space.
invented entities (1)
  • Completely irreducible generalized Bratteli diagram no independent evidence
    purpose: To identify diagrams that are isomorphic exclusively to irreducible ones
    Newly defined in the paper as a stronger form of irreducibility.

pith-pipeline@v0.9.0 · 5608 in / 1214 out tokens · 78373 ms · 2026-05-20T00:04:32.243347+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

207 extracted references · 207 canonical work pages

  1. [1]

    Shimomura, Takashi , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.1090/proc/13575 , URL =

    work page doi:10.1090/proc/13575 2017

  2. [2]

    1997 , PAGES =

    Aaronson, Jon , TITLE =. 1997 , PAGES =. doi:10.1090/surv/050 , URL =

    work page doi:10.1090/surv/050 1997

  3. [3]

    Akin, Ethan , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2005 , NUMBER =. doi:10.1090/S0002-9947-04-03524-X , URL =

    work page doi:10.1090/s0002-9947-04-03524-x 2005

  4. [4]

    Vershik, A. M. , TITLE =. Funktsional. Anal. i Prilozhen. , FJOURNAL =. 2011 , NUMBER =

    work page 2011

  5. [5]

    Vershik, A. M. , TITLE =. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , FJOURNAL =. 2014 , NUMBER =

    work page 2014

  6. [6]

    and Bezuglyi, S

    Adamska, M. and Bezuglyi, S. and Karpel, O. and Kwiatkowski, J. , title=. Ergodic Theory Dynam. Systems , FJOURNAL =. 2017 , NUMBER =. doi:doi:10.1017/etds.2016.8 , URL =

    work page doi:10.1017/etds.2016.8 2017

  7. [7]

    and Golestani, Nasser , TITLE =

    Amini, Massoud and Elliott, George A. and Golestani, Nasser , TITLE =. Canad. J. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.4153/S0008414X19000452 , URL =

    work page doi:10.4153/s0008414x19000452 2021

  8. [8]

    Avni, Nir and Breuer, Jonathan and Simon, Barry , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.aim.2020.107241 , URL =

    work page doi:10.1016/j.aim.2020.107241 2020

  9. [9]

    2021 , month =

    Beltr\'an, Elmer R and Bissacot, Rodrigo and Endo, Eric O , title =. 2021 , month =. doi:10.1088/1361-6544/abf84d , url =

    work page doi:10.1088/1361-6544/abf84d 2021

  10. [10]

    2016 , PAGES =

    Combinatorics, words and symbolic dynamics , SERIES =. 2016 , PAGES =. doi:10.1017/CBO9781139924733 , URL =

    work page doi:10.1017/cbo9781139924733 2016

  11. [11]

    Bezuglyi, Sergey and Jorgensen, Palle E. T. , TITLE =. J. Fourier Anal. Appl. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00041-021-09827-0 , URL =

    work page doi:10.1007/s00041-021-09827-0 2021

  12. [12]

    Bezuglyi, Sergey and Kwiatkowski, Jan , TITLE =. Topol. Methods Nonlinear Anal. , FJOURNAL =. 2000 , NUMBER =. doi:10.12775/TMNA.2000.046 , URL =

    work page doi:10.12775/tmna.2000.046 2000

  13. [13]

    and Dooley, A

    Bezuglyi, S. and Dooley, A. H. and Medynets, K. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2005 , NUMBER =. doi:10.1090/S0002-9939-05-07777-4 , URL =

    work page doi:10.1090/s0002-9939-05-07777-4 2005

  14. [14]

    and Dooley, A

    Bezuglyi, S. and Dooley, A. H. and Kwiatkowski, J. , TITLE =. Topol. Methods Nonlinear Anal. , FJOURNAL =. 2006 , NUMBER =

    work page 2006

  15. [15]

    arXiv:2404.14654 , YEAR =

    Bezuglyi, Sergey and Karpel, Olena and Kwiatkowski, Jan and Wata, Marcin , TITLE =. arXiv:2404.14654 , YEAR =

    work page arXiv

  16. [16]

    and Karpel, Olena and Kwiatkowski, Jan , TITLE =

    Bezuglyi, Sergey and Jorgensen, Palle E.T. and Karpel, Olena and Kwiatkowski, Jan , TITLE =. Fund. Math. , FJOURNAL =. 2025 , NUMBER =. doi:10.4064/fm240916-6-6 , URL =

    work page doi:10.4064/fm240916-6-6 2025

  17. [17]

    Bezuglyi, Sergey and Dudko, Artem and Karpel, Olena , TITLE =. J. Math. Phys. Anal. Geom. , FJOURNAL =. 2026 , NUMBER =

    work page 2026

  18. [18]

    Bezuglyi, Sergey and Karpel, Olena and Kwiatkowski, Jan , TITLE =. J. Math. Anal. Appl. , FJOURNAL =. 2019 , NUMBER =. doi:10.1016/j.jmaa.2019.123431 , URL =

    work page doi:10.1016/j.jmaa.2019.123431 2019

  19. [19]

    Bezuglyi, Sergey and Karpel, Olena and Kwiatkowski, Jan , TITLE =. J. Math. Phys. Anal. Geom. , FJOURNAL =. 2024 , NUMBER =. doi:10.15407/mag20.01.003 , URL =

    work page doi:10.15407/mag20.01.003 2024

  20. [20]

    and Medynets, K

    Bezuglyi, S. and Medynets, K. , TITLE =. Colloq. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.4064/cm110-2-6 , URL =

    work page doi:10.4064/cm110-2-6 2008

  21. [21]

    and Kwiatkowski, J

    Bezuglyi, S. and Kwiatkowski, J. and Medynets, K. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2009 , NUMBER =. doi:10.1017/S0143385708000230 , URL =

    work page doi:10.1017/s0143385708000230 2009

  22. [22]

    and Kwiatkowski, J

    Bezuglyi, S. and Kwiatkowski, J. and Medynets, K. and Solomyak, B. , Date-Added =. Invariant measures on stationary. Ergodic Theory Dynam. Systems , Mrclass =. 2010 , Bdsk-Url-1 =. doi:10.1017/S0143385709000443 , Fjournal =

    work page doi:10.1017/s0143385709000443 2010

  23. [23]

    and Karpel, O

    Bezuglyi, S. and Karpel, O. , TITLE =. J. Funct. Anal. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.jfa.2011.08.009 , URL =

    work page doi:10.1016/j.jfa.2011.08.009 2011

  24. [24]

    and Kwiatkowski, J

    Bezuglyi, S. and Kwiatkowski, J. and Medynets, K. and Solomyak, B. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2013 , NUMBER =. doi:10.1090/S0002-9947-2012-05744-8 , URL =

    work page doi:10.1090/s0002-9947-2012-05744-8 2013

  25. [25]

    and Kwiatkowski, J

    Bezuglyi, S. and Kwiatkowski, J. and Yassawi, R. , TITLE =. Canad. J. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.4153/CJM-2013-041-6 , URL =

    work page doi:10.4153/cjm-2013-041-6 2014

  26. [26]

    and Handelman, D

    Bezuglyi, S. and Handelman, D. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2014 , NUMBER =. doi:10.1090/S0002-9947-2014-06035-2 , URL =

    work page doi:10.1090/s0002-9947-2014-06035-2 2014

  27. [27]

    and Jorgensen, Palle E

    Bezuglyi, S. and Jorgensen, Palle E. T. , TITLE =. Trends in harmonic analysis and its applications , SERIES =. 2015 , MRCLASS =. doi:10.1090/conm/650/13008 , URL =

    work page doi:10.1090/conm/650/13008 2015

  28. [28]

    and Karpel, O

    Bezuglyi, S. and Karpel, O. , TITLE =. Dynamics and numbers , SERIES =. 2016 , MRCLASS =. doi:10.1090/conm/669/13421 , URL =

    work page doi:10.1090/conm/669/13421 2016

  29. [29]

    and Karpel, O

    Bezuglyi, S. and Karpel, O. and Kwiatkowski, J. , TITLE =. Zh. Mat. Fiz. Anal. Geom. , FJOURNAL =. 2015 , NUMBER =. doi:10.15407/mag11.01.003 , URL =

    work page doi:10.15407/mag11.01.003 2015

  30. [30]

    Bezuglyi, Sergey and Yassawi, Reem , TITLE =. Dyn. Syst. , FJOURNAL =. 2017 , NUMBER =. doi:10.1080/14689367.2016.1197888 , URL =

    work page doi:10.1080/14689367.2016.1197888 2017

  31. [31]

    Dynamics: topology and numbers , SERIES =

    Bezuglyi, Sergey and Karpel, Olena , TITLE =. Dynamics: topology and numbers , SERIES =. [2020] 2020 , MRCLASS =. doi:10.1090/conm/744/14988 , URL =

    work page doi:10.1090/conm/744/14988 2020

  32. [32]

    Bezuglyi, Sergey and Jorgensen, Palle E. T. , TITLE =. Dissertationes Math. , FJOURNAL =. 2022 , PAGES =. doi:10.4064/dm826-12-2021 , URL =

    work page doi:10.4064/dm826-12-2021 2022

  33. [33]

    Bezuglyi, Sergey and Jorgensen, Palle E. T. , keywords =. 2022 , copyright =. doi:10.48550/ARXIV.2210.14059 , url =

    work page doi:10.48550/arxiv.2210.14059 2022

  34. [34]

    and Exel, R

    Bissacot, R. and Exel, R. and Frausino, R. and Raszeja, T. , keywords =. Thermodynamic Formalism for Generalized. 2022 , doi =

    work page 2022

  35. [35]

    Entropy , FJOURNAL =

    Bobok, Jozef and Bruin, Henk , TITLE =. Entropy , FJOURNAL =. 2016 , NUMBER =. doi:10.3390/e18060234 , URL =

    work page doi:10.3390/e18060234 2016

  36. [36]

    Boshernitzan, Michael , TITLE =. J. Analyse Math. , FJOURNAL =. 1984 , PAGES =. doi:10.1007/BF02790191 , URL =

    work page doi:10.1007/bf02790191 1984

  37. [37]

    Duke Math

    Boshernitzan, Michael , TITLE =. Duke Math. J. , FJOURNAL =. 1985 , NUMBER =. doi:10.1215/S0012-7094-85-05238-X , URL =

    work page doi:10.1215/s0012-7094-85-05238-x 1985

  38. [38]

    , TITLE =

    Boshernitzan, Michael D. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1992 , NUMBER =. doi:10.1017/S0143385700006866 , URL =

    work page doi:10.1017/s0143385700006866 1992

  39. [39]

    Ergodic Theory Dynam

    Boyle, Mike , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1983 , NUMBER =. doi:10.1017/S0143385700002133 , URL =

    work page doi:10.1017/s0143385700002133 1983

  40. [40]

    , TITLE =

    Bratteli, O. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1972 , PAGES =

    work page 1972

  41. [41]

    Bratteli, Ola and Jorgensen, Palle E. T. and Kim, Ki Hang and Roush, Fred , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2002 , NUMBER =. doi:10.1017/S0143385702000044 , URL =

    work page doi:10.1017/s0143385702000044 2002

  42. [42]

    Bratteli, Ola and Jorgensen, Palle E. T. and Kim, Ki Hang and Roush, Fred , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2001 , NUMBER =. doi:10.1017/S014338570100178X , URL =

    work page doi:10.1017/s014338570100178x 2001

  43. [43]

    Bratteli, Ola and Jorgensen, Palle E. T. and Kim, Ki Hang and Roush, Fred , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2000 , NUMBER =. doi:10.1017/S0143385700000912 , URL =

    work page doi:10.1017/s0143385700000912 2000

  44. [44]

    Ergodic Theory Dynam

    Bressaud, Xavier and Durand, Fabien and Maass, Alejandro , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2010 , NUMBER =. doi:10.1017/S0143385709000236 , URL =

    work page doi:10.1017/s0143385709000236 2010

  45. [45]

    and Kwiatkowski, J

    Bulatek, W. and Kwiatkowski, J. , TITLE =. Publ. Mat. , FJOURNAL =. 1990 , PAGES =

    work page 1990

  46. [46]

    Developments in language theory,

    Cassaigne, Julien , TITLE =. Developments in language theory,. 1996 , MRCLASS =

    work page 1996

  47. [47]

    Chaika, Jon and Masur, Howard , TITLE =. J. Mod. Dyn. , FJOURNAL =. 2015 , PAGES =. doi:10.3934/jmd.2015.9.289 , URL =

    work page doi:10.3934/jmd.2015.9.289 2015

  48. [48]

    and Simon, Barry and Zinchenko, Maxim , TITLE =

    Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim , TITLE =. Spectral analysis, differential equations and mathematical physics: a festschrift in honor of. 2013 , MRCLASS =. doi:10.1090/pspum/087/01429 , URL =

    work page doi:10.1090/pspum/087/01429 2013

  49. [49]

    and Simon, Barry and Zinchenko, Maxim , TITLE =

    Christiansen, Jacob S. and Simon, Barry and Zinchenko, Maxim , TITLE =. Constr. Approx. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00365-012-9152-4 , URL =

    work page doi:10.1007/s00365-012-9152-4 2012

  50. [50]

    Quasi-stationary distributions , SERIES =

    Collet, Pierre and Mart\'. Quasi-stationary distributions , SERIES =. 2013 , PAGES =. doi:10.1007/978-3-642-33131-2 , URL =

    work page doi:10.1007/978-3-642-33131-2 2013

  51. [51]

    and Feldman, J

    Connes, A. and Feldman, J. and Weiss, B. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1981 , NUMBER =. doi:10.1017/s014338570000136x , URL =

    work page doi:10.1017/s014338570000136x 1981

  52. [52]

    1994 , PAGES =

    Connes, Alain , TITLE =. 1994 , PAGES =

    work page 1994

  53. [53]

    Cornfeld, I. P. and Fomin, S. V. and Sina. Ergodic theory , SERIES =. 1982 , PAGES =. doi:10.1007/978-1-4615-6927-5 , URL =

    work page doi:10.1007/978-1-4615-6927-5 1982

  54. [54]

    Ergodic Theory Dynam

    Cortez, Maria Isabel , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2006 , NUMBER =. doi:10.1017/S0143385706000319 , URL =

    work page doi:10.1017/s0143385706000319 2006

  55. [55]

    Cortez, Maria Isabel and Durand, Fabien and Host, Bernard and Maass, Alejandro , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 2003 , NUMBER =. doi:10.1112/S0024610703004320 , URL =

    work page doi:10.1112/s0024610703004320 2003

  56. [56]

    and Hedlund, G

    Coven, Ethan M. and Hedlund, G. A. , TITLE =. Math. Systems Theory , FJOURNAL =. 1973 , PAGES =. doi:10.1007/BF01762232 , URL =

    work page doi:10.1007/bf01762232 1973

  57. [57]

    Cyr, Van and Kra, Bryna , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2019 , NUMBER =. doi:10.4171/JEMS/838 , URL =

    work page doi:10.4171/jems/838 2019

  58. [58]

    Ergodic Theory Dynam

    Damron, Michael and Fickenscher, Jon , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2017 , NUMBER =. doi:10.1017/etds.2015.138 , URL =

    work page doi:10.1017/etds.2015.138 2017

  59. [59]

    Preprint , FJOURNAL =

    Damron, Michael and Fickenscher, Jon , TITLE =. Preprint , FJOURNAL =

    work page

  60. [60]

    and Silva, Cesar E

    Danilenko, Alexandre I. and Silva, Cesar E. , TITLE =. Mathematics of complexity and dynamical systems. 2012 , MRCLASS =. doi:10.1007/978-1-4614-1806-1_22 , URL =

    work page doi:10.1007/978-1-4614-1806-1_22 2012

  61. [61]

    Donoso, Sebasti\'an and Durand, Fabien and Maass, Alejandro and Petite, Samuel , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/tran/8315 , URL =

    work page doi:10.1090/tran/8315 2021

  62. [62]

    and Jackson, S

    Dougherty, R. and Jackson, S. and Kechris, A. S. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1994 , NUMBER =. doi:10.2307/2154620 , URL =

    work page doi:10.2307/2154620 1994

  63. [63]

    , TITLE =

    Downarowicz, T. , TITLE =. Bull. Polish Acad. Sci. , FJOURNAL =. 1990 , NUMBER =. doi:, URL =

    work page 1990

  64. [64]

    , TITLE =

    Downarowicz, T. , TITLE =. Israel J. Math. , FJOURNAL =. 1991 , NUMBER =. doi:10.1007/BF02775789 , URL =

    work page doi:10.1007/bf02775789 1991

  65. [65]

    , TITLE =

    Downarowicz, T. , TITLE =. Israel J. Math. , FJOURNAL =. 2006 , PAGES =. doi:10.1007/BF02773826 , URL =

    work page doi:10.1007/bf02773826 2006

  66. [66]

    , TITLE =

    Downarowicz, T. , TITLE =. Israel J. Math. , FJOURNAL =. 2008 , PAGES =. doi:10.1007/s11856-008-1009-y , URL =

    work page doi:10.1007/s11856-008-1009-y 2008

  67. [67]

    and Kwiatkowski, J

    Downarowicz, T. and Kwiatkowski, J. and Lacroix, Y. , TITLE =. Colloq. Math. , FJOURNAL =. 1995 , PAGES =. doi:, URL =

    work page 1995

  68. [68]

    and Maass, A

    Downarowicz, T. and Maass, A. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2008 , PAGES =. doi:, URL =

    work page 2008

  69. [69]

    Discrete Contin

    Downarowicz, Tomasz and Karpel, Olena , TITLE =. Discrete Contin. Dyn. Syst. , FJOURNAL =. 2018 , NUMBER =. doi:10.3934/dcds.2018044 , URL =

    work page doi:10.3934/dcds.2018044 2018

  70. [70]

    Studia Math

    Downarowicz, Tomasz and Karpel, Olena , TITLE =. Studia Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.4064/sm170519-5-2 , URL =

    work page doi:10.4064/sm170519-5-2 2019

  71. [71]

    Discrete Math

    Durand, Fabien , TITLE =. Discrete Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1016/S0012-365X(97)00029-0 , URL =

    work page doi:10.1016/s0012-365x(97)00029-0 1998

  72. [72]

    Invariant measures for substitutions on countable alphabets , JOURNAL =

    Domingos, Weberty and Ferenczi, S\'. Invariant measures for substitutions on countable alphabets , JOURNAL =. 2024 , NUMBER =. doi:10.1017/etds.2023.113 , URL =

    work page doi:10.1017/etds.2023.113 2024

  73. [73]

    and Host, B

    Durand, F. and Host, B. and Skau, C. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1999 , NUMBER =. doi:10.1017/S0143385799133947 , URL =

    work page doi:10.1017/s0143385799133947 1999

  74. [74]

    Ergodic Theory Dynam

    Durand, Fabien , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2000 , NUMBER =. doi:10.1017/S0143385700000584 , URL =

    work page doi:10.1017/s0143385700000584 2000

  75. [75]

    Combinatorics on

    Durand, Fabien , Booktitle =. Combinatorics on

    work page

  76. [76]

    Durand, Fabien and Frank, Alexander and Maass, Alejandro , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2019 , NUMBER =. doi:10.4171/JEMS/849 , URL =

    work page doi:10.4171/jems/849 2019

  77. [77]

    2022 , PAGES =

    Durand, Fabien and Perrin, Dominique , TITLE =. 2022 , PAGES =. doi:10.1017/9781108976039 , URL =

    work page doi:10.1017/9781108976039 2022

  78. [78]

    Dye, H. A. , TITLE =. Amer. J. Math. , FJOURNAL =. 1959 , PAGES =. doi:10.2307/2372852 , URL =

    work page doi:10.2307/2372852 1959

  79. [79]

    Dye, H. A. , TITLE =. Amer. J. Math. , FJOURNAL =. 1963 , PAGES =. doi:10.2307/2373108 , URL =

    work page doi:10.2307/2373108 1963

  80. [80]

    , TITLE =

    Effros, Edward G. , TITLE =. 1981 , PAGES =

    work page 1981

Showing first 80 references.