Residual entropy, conditional entropy and subshift covers.

*(English)*Zbl 1030.37012This paper addresses the existence and entropy of symbolic extensions of a topological dynamical system \((X,T)\), topological conditional entropy of a factor map, asymptotic \(h\)-expansiveness, and some other related issues. The key notions are: subshift cover \((Y,S)\), i.e., a topological extension \(\phi:(Y,S)\to (X,T)\), where \((Y,S)\) has the form of a symbolic system (subshift over a finite alphabet), and residual entropy \(h_{\text{res}}(T)\) defined as the infimum of the topological entropies of all possible subshift covers \((Y,S)\) minus the topological entropy of \(T\). If no subshift covers of \(T\) exist, we set \(h_{res}(T)=\infty\).

Crucial for the development of the theory of symbolic extensions is Example 3.1 in the paper, of a finite entropy system \((X,T)\) admitting no symbolic extensions at all. This example settles a corresponding question raised by J. Auslander in the late 1980’s.

The authors introduce a notion of (topological) conditional entropy of a quotient map \(\phi:(Y,S)\to(X,T)\) as \(e^*(\phi):=\inf_{\mathcal V}h(S|\phi^{-1}\mathcal V)\), where \(\mathcal V\) ranges over all (finite) open covers of \(X\) and \(h(S|\mathcal U)\) is the topological conditional entropy of \(S\) given an open cover \(\mathcal U\), as defined classically by M. Misiurewicz [Stud. Math. 66, 175–200 (1976; Zbl 0355.54035)]. Following several inequalities connecting their new notion with the Misiurewicz’ topological conditional (tail) entropy \(h^*\) of both \(T\) and \(S\), the authors prove a variational principle for their notion: \[ e^*(\phi)\geq \sup_m(h_m(S) - h_{\phi(m)}(T))\geq e^*(\phi) - h^*(T) \] (\(m\) ranges over invarint measures on \(Y\)), in particular, \(e^*(\phi) = \sup_m(h_m(S) - h_{\phi(m)}(T))\) for \(T\) asymptotically \(h\)-expansive (i.e., when \(h^*(T)=0\); see also a work by J. Serafin and the reviewer [Fundam. Math. 172, 217–247 (2002; Zbl 1115.37308)] for a similar result without the restriction, involving a slightly different notion).

An important Theorem 7.4 says that asymptotically \(h\)-expansive systems \((X,T)\) have zero residual entropy, even more – there exists a subshift cover \(\phi\) with \(e^*(\phi) = 0\) (i.e., by the variational principle, such that \(\phi\) preserves entropy of all invariant measures). Combined with a remarkable theorem by J. Buzzi [Isr. J. Math. 100, 125–161 (1997; Zbl 0889.28009)] which asserts that every \(C^\infty\) diffeomorphism of a compact Riemannian manifold is asymptotically \(h\)-expansive, this leads to a striking corollary (Theorem 7.8), which may be considered one of the most important facts in the theory of symbolic extensions, and which can be phrased as follows:

Every \(C^\infty\) diffeomorphism of a compact Riemannian manifold has a symbolic extension which preserves entropy of all invariant measures.

The authors (implicitly) pose a question whether positive residual entropy is possible in \(C^r\) systems \(r<\infty\). This problem has been recently solved affirmatively by S. Newhouse and the reviewer [Symbolic extensions and smooth dynamical systems, Invent. Math. 160, No. 3, 453–499 (2005; Zbl 1067.37018)].

In dimension zero there exists a formula for residual entropy obtained by the reviewer [Ergodic Theory Dyn. Syst. 21, 1051–1070 (2001; Zbl 1055.37022)] referring to measure theoretic entropies. In the paper under review a different formula is given, in terms of certain functions on blocks called “word oracle”. It is worth mentioning that various characterizations of residual entropy in the general case (no dimension restrictions) using languages of both measure theoretic entropies and word oracles have been recently obtained by the first author and the reviewer [The entropy theory of symbolic extensions, Invent. Math. 156, No. 1, 119–161 (2004; Zbl 1216.37004)], and, by the reviewer, in the language of spanning and separated sets, \(\varepsilon\)-orbits, and many other classical terms [Entropy structure, J. Anal. Math. 96, 57–116 (2005; Zbl 1151.37020)].

The paper is concluded by several appendices containing constructions of zero-dimensional covers of certain systems and an example of an infinite residual entropy system on a surface.

Crucial for the development of the theory of symbolic extensions is Example 3.1 in the paper, of a finite entropy system \((X,T)\) admitting no symbolic extensions at all. This example settles a corresponding question raised by J. Auslander in the late 1980’s.

The authors introduce a notion of (topological) conditional entropy of a quotient map \(\phi:(Y,S)\to(X,T)\) as \(e^*(\phi):=\inf_{\mathcal V}h(S|\phi^{-1}\mathcal V)\), where \(\mathcal V\) ranges over all (finite) open covers of \(X\) and \(h(S|\mathcal U)\) is the topological conditional entropy of \(S\) given an open cover \(\mathcal U\), as defined classically by M. Misiurewicz [Stud. Math. 66, 175–200 (1976; Zbl 0355.54035)]. Following several inequalities connecting their new notion with the Misiurewicz’ topological conditional (tail) entropy \(h^*\) of both \(T\) and \(S\), the authors prove a variational principle for their notion: \[ e^*(\phi)\geq \sup_m(h_m(S) - h_{\phi(m)}(T))\geq e^*(\phi) - h^*(T) \] (\(m\) ranges over invarint measures on \(Y\)), in particular, \(e^*(\phi) = \sup_m(h_m(S) - h_{\phi(m)}(T))\) for \(T\) asymptotically \(h\)-expansive (i.e., when \(h^*(T)=0\); see also a work by J. Serafin and the reviewer [Fundam. Math. 172, 217–247 (2002; Zbl 1115.37308)] for a similar result without the restriction, involving a slightly different notion).

An important Theorem 7.4 says that asymptotically \(h\)-expansive systems \((X,T)\) have zero residual entropy, even more – there exists a subshift cover \(\phi\) with \(e^*(\phi) = 0\) (i.e., by the variational principle, such that \(\phi\) preserves entropy of all invariant measures). Combined with a remarkable theorem by J. Buzzi [Isr. J. Math. 100, 125–161 (1997; Zbl 0889.28009)] which asserts that every \(C^\infty\) diffeomorphism of a compact Riemannian manifold is asymptotically \(h\)-expansive, this leads to a striking corollary (Theorem 7.8), which may be considered one of the most important facts in the theory of symbolic extensions, and which can be phrased as follows:

Every \(C^\infty\) diffeomorphism of a compact Riemannian manifold has a symbolic extension which preserves entropy of all invariant measures.

The authors (implicitly) pose a question whether positive residual entropy is possible in \(C^r\) systems \(r<\infty\). This problem has been recently solved affirmatively by S. Newhouse and the reviewer [Symbolic extensions and smooth dynamical systems, Invent. Math. 160, No. 3, 453–499 (2005; Zbl 1067.37018)].

In dimension zero there exists a formula for residual entropy obtained by the reviewer [Ergodic Theory Dyn. Syst. 21, 1051–1070 (2001; Zbl 1055.37022)] referring to measure theoretic entropies. In the paper under review a different formula is given, in terms of certain functions on blocks called “word oracle”. It is worth mentioning that various characterizations of residual entropy in the general case (no dimension restrictions) using languages of both measure theoretic entropies and word oracles have been recently obtained by the first author and the reviewer [The entropy theory of symbolic extensions, Invent. Math. 156, No. 1, 119–161 (2004; Zbl 1216.37004)], and, by the reviewer, in the language of spanning and separated sets, \(\varepsilon\)-orbits, and many other classical terms [Entropy structure, J. Anal. Math. 96, 57–116 (2005; Zbl 1151.37020)].

The paper is concluded by several appendices containing constructions of zero-dimensional covers of certain systems and an example of an infinite residual entropy system on a surface.

Reviewer: Tomasz Downarowicz (Wrocław)

##### MSC:

37B40 | Topological entropy |

37B10 | Symbolic dynamics |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

##### Keywords:

residual entropy; symbolic extension; topological conditional entropy; asymptotic \(h\)-expansiveness
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\textit{M. Boyle} et al., Forum Math. 14, No. 5, 713--757 (2002; Zbl 1030.37012)

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