Skip to main content
SpringerLink
Log in

Uniform Generators, Symbolic Extensions with an Embedding, and Structure of Periodic Orbits

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

For a topological dynamical system \((X,T)\) we define a uniform generator as a finite measurable partition such that the symmetric cylinder sets in the generated process shrink in diameter uniformly to zero. The problem of existence and optimal cardinality of uniform generators has lead us to new challenges in the theory of symbolic extensions. At the beginning we show that uniform generators can be identified with so-called symbolic extensions with an embedding, i.e., symbolic extensions admitting an equivariant measurable selector from preimages. From here we focus on such extensions and we strive to characterize the collection of the corresponding extension entropy functions on invariant measures. For aperiodic zero-dimensional systems we show that this collection coincides with that of extension entropy functions in arbitrary symbolic extensions, which, by the general theory of symbolic extensions, is known to coincide with the collection of all affine superenvelopes of the entropy structure of the system. In particular, we recover, after Burguet (Monatsh Math 184:21–49, 2017), that an aperiodic zero-dimensional system is asymptotically h-expansive if and only if it admits an isomorphic symbolic extension. Next we pass to systems with periodic points, and we introduce the notion of a period tail structure, which captures the local growth rate of periodic orbits. Finally, we succeed in precisely identifying the wanted collection of extension entropy functions in symbolic extensions with an embedding: these are all the affine superenvelopes of the usual entropy structure which lie above certain threshold function determined by the period tail structure. This characterization allows us, among other things, to give estimates (and in examples to compute precisely) of the optimal cardinality of a uniform generator. As a byproduct, we prove a theorem saying that every zero-dimensional system admits an aperiodic zero-dimensional extension which is isomorphic on aperiodic measures and otherwise principal (periodic measures lift to measures of entropy zero).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. See Remark 1.3 for the precise meaning of measurability.

  2. Later we will show that the measurability assumption of \(\mathcal {P}\) can be dropped. That is, the existence of a “non-measurable uniform generator” implies the existence of a measurable one—see Remark 4.15.

  3. The cited paper is the first work dealing with the subject of symbolic extensions with an embedding. For asymptotically h-expansive systems the existence of an isomorphic extension is proved even in presence of periodic points as long as they satisfy a condition called asymptotic per-expansiveness.

  4. In many situations, the equality \(E(\mu ) = h(\mu )\) holds on a large set of invariant measures. For instance, it is known that \(E_{\mathsf {min}}= h\) on a residual subset of \(\mathcal {M}_T(X)\) and whenever \(E_{\mathsf {min}}\) is affine (which is always the case for example whenever the set of ergodic measures is closed), then it is the most natural choice for E. More details can be found in Sections 3 and 4 of [2].

  5. Throughout this paper we calculate all entropies using the logarithm to base 2 (we will write just “\(\log \)”, but in most cases we will not simplify \(\log 2\)).

  6. Recently, M. Hochman extended the result also to systems with periodic points [9].

  7. Technically, we should bound the ratios \(\frac{p^{\mathsf {min}}_{k+1}}{p^{\mathsf {max}}_k}\), but since we always have \(p^{\mathsf {max}}_k\le 3p^{\mathsf {min}}_k\), it suffices to bound the ratios \(\frac{p^{\mathsf {min}}_{k+1}}{p^{\mathsf {min}}_k}\).

  8. A point is generic for an invariant measure \(\mu \) if the empirical measures along its orbit converge weakly-star to \(\mu \). In symbolic systems, equivalently: the density of occurrences of every block equals its measure. For \(\mu \) ergodic the set of generic points has measure 1.

  9. For instance, a system which has k fixpoints cannot be embedded in a subshift over less than k symbols, but otherwise the symbolic extension may have, for example, zero entropy and, except in the fixpoints, use, say, only two symbols.

  10. We recall that a subset \(A\subset X\) of a topological dynamical system \((X,T)\) is a null set if it has measure zero for all \(\mu \in \mathcal {M}_T(X)\).

  11. Notice that we do not require \(\pi (X_0)\) to have almost null complement in \(X'\).

  12. Since we do not require that the corresponding marker sets are nested, a k-rectangle is enclosed by markers only in row k, not in every row 1 through k like on the Fig. 1.

  13. This practically means that either there already are some markers (Krieger’s or periodic put in a preceding step) appearing one every p positions, and then we do not put any markers in this step, or there are only Krieger’s markers outside the long k-rectangle and we need to mind only the two closest external ones.

  14. Although the marker symbols do not allow to distinguish between Krieger’s, periodic and stretched markers, we can always determine Krieger’s markers by removing all markers and repeating the first (continuous) algorithm.

  15. This is the place in the proof where it becomes essential that \((\breve{Y},\breve{S})\) extends not just \((X,T)\) but also the enhanced system \((\hat{X},\hat{T})\). By referring to the topological entropies of the subsystems \(\mathbf {S}\subset \mathbf {S}_n\) we implicitly involve some “local” limit periodic capacities of \((X,T)\).

  16. Further, we will refer to such a prolongation as harmonic. Harmonic prolongations are harmonic functions, i.e., respect integral averaging, which is in general a stronger condition than just being affine. However, upper semicontinuous affine functions are harmonic.

  17. In order for a symbolic extension with partial embedding to exist, we must assume that the “partial” supremum periodic capacity \(\sup _n\frac{1}{n}\log (\#\mathsf {Per}^*_n)\) is finite. This enables us to apply the Lebesgue Dominated Convergence Theorem.

  18. Equality holds when the sequence \(\{\theta _k\}\) has so-called order of accumulation 1. In general, the smallest repair function is obtained by repeating an iterative procedure as many times as the order of accumulation, which is always a countable ordinal. See Chapter 8 in [7] for more details.

  19. For the harmonic prolongation only the values at extreme points matter and the domain of \(\hat{u}\) contains all ergodic measures of \((\hat{X},\hat{T})\).

  20. Tail entropy is known mainly under the confusing name of “topological conditional entropy”. It turns out (see [6]) that \(h^*(X,T)\) is equal to \(\sup u^{\mathcal {H}}_1\), where \(u^{\mathcal {H}}_1\) is computed for the tails of the entropy structure as \(u^{\mathcal {H}}_1=\lim _k(\widetilde{h-h_k})\).

References

  1. Boyle, M.: Lower entropy factors of sofic systems. Ergod. Theory Dyn. Syst. 3, 541–557 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boyle, M., Downarowicz, T.: The entropy theory of symbolic extensions. Invent. Math. 156, 119–161 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burguet, D.: Embedding asymptotically expansive systems. Monatsh. Math. 184, 21–49 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burguet, D.: Periodic expansiveness of smooth surface diffeomorphisms and applications. J. Eur. Math. Soc. (to appear)

  5. Buzzi, J.: Intrinsic ergodicity for smooth interval maps. Israel J. Math. 100, 125–161 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Downarowicz, T.: Entropy structure. J. Anal. Math. 96, 57–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Downarowicz, T.: Entropy in Dynamical Systems, New Mathematical Monographs, vol. 18. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  8. Hochman, M.: Isomorphism and embedding into Markov shifts off universally null sets. Acta Appl. Math. 126(1), 187–201 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hochman, M.: Every Borel automorphism without finite invariant measures admits a two-set generator (preprint) arXiv:1508.02335

  10. Krieger, W.: On the subsystems of topological Markov chains. Ergod. Theory. Dyn. Syst. 2, 195–202 (1882)

    MathSciNet  MATH  Google Scholar 

  11. Kulesza, J.: Zero-dimensional covers of finite-dimensional dynamical systems. Ergod. Theory. Dyn. Syst. 15, 939–950 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lindenstrauss, E.: Mean dimension, small entropy factors, and an embedding theorem. Publ. Math. IHES 89, 227–262 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Misiurewicz, M.: Topological conditional entropy. Stud. Math. 55, 175–200 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ornstein, D. S.: Ergodic Theory, Randomness, and Dynamical Systems, James K. Whittemore Lectures in Mathematics Given at Yale University, Yale Mathematical Monographs, No. 5, Yale University Press, New Haven, Conn.-London (1974)

  15. Serafin, J.: Universally finitary symbolic extensions. Fund. Math. 206, 281–285 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Serafin, J.: A faithful symbolic extension. Commun. Pure Appl. Anal. 11, 1051–1062 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shields, P.: The ergodic theory of discrete sample paths, vol. 13. AMS Graduate Studies in Mathematics (1996)

Download references

Author information

Authors and Affiliations

  1. CNRS, Laboratoire de Probabilités, Statistique et Modélisation, LPSM, Sorbonne Université, 75005, Paris, France

    David Burguet

  2. Faculty of Pure and Applied Mathematics, Wrocław University of Technology, 50-370, Wrocław, Poland

    Tomasz Downarowicz

Authors
  1. David Burguet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomasz Downarowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Burguet.

Additional information

The research of the second author is supported by the NCN (National Science Center, Poland) Grant 2013/08/A/ST1/00275.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burguet, D., Downarowicz, T. Uniform Generators, Symbolic Extensions with an Embedding, and Structure of Periodic Orbits. J Dyn Diff Equat 31, 815–852 (2019). https://doi.org/10.1007/s10884-018-9674-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-018-9674-y

Keywords

Search

Navigation