## A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems (1996)

Venue: | Ergodic Theory Dynam. Systems |

Citations: | 29 - 16 self |

### BibTeX

@ARTICLE{Barreira96anon-additive,

author = {Luis M. Barreira},

title = {A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems},

journal = {Ergodic Theory Dynam. Systems },

year = {1996},

volume = {16},

number = {5},

pages = {871--927}

}

### Years of Citing Articles

### OpenURL

### Abstract

A non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they are coded by an arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

### Citations

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Citation Context ...ver, one can show that on suciently small rectangles the map (x; y) 7! [x; y] is a C homeomorphism with C inverse if and only if the holonomy maps are C (apply for example Proposition 19.1.1 in [KH]). This implies the following. Corollary 3.19. Let be a basic set of a topologically mixing C 1 Axiom A dieomorphism. Then, for all suciently small 2 (0; 1), we have sup x2 r u (x) + r s (x) ... |

541 |
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Citation Context ...wing limits exist: PZ () def = lim jUj!0 PZ (; U); CP Z () def = lim jUj!0 CP Z (; U); CP Z () def = lim jUj!0 CP Z (; U): Proof. This is a slight modication of the proof of Proposition 2.8 in [Bo2]. Let V be asnite open cover of X with diameter smaller than the Lebesgue number of U. Each element V 2 V is contained in some element U(V ) 2 U. We write U(V) = U(V 1 ) U(V n ) for each V 2 W n... |

110 |
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Citation Context ...ted in Example 3.12 with the limit set of a geometric construction in R 2 . Hence, Example 3.12 exhibits an induced map on a limit set that is asymptotically conformal (and even weakly conformal; see =-=[Pe2] fo-=-r the denition), but not conformal. 3.2. Basic sets of Axiom A ] homeomorphisms. 3.2.1. Denitions. Let f : X ! X be a homeomorphism of the compact metric space (X; d). Given x 2 X and " > 0, we d... |

106 |
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Citation Context ...]. In addition, we establish thesrst general dimension estimates for repellers of C 1 expanding maps that may not be conformal. Our methods of proof were inspired in the proofs of Palis and Takens in =-=[P-=-T] of results of Takens for dynamically dened Cantor sets. In the case of invertible maps, we recover the formula giving the Hausdor di4 LUIS M. BARREIRA mension of basic sets of saddle-type Axiom A s... |

51 |
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Citation Context ...n. The quantity h f (Z) = PZ (0) coincides with the notion of topological entropy for non-compact sets introduced by Pesin and Pitskel' in [PP], and is equivalent to the notion introduced by Bowen in =-=[Bo1]-=- (see [PP]). The quantities Ch f (Z) = CP Z (0) and Ch f (Z) = CP Z (0) coincide with the notions of lower and upper capacity topological entropies introduced by Pesin in [Pe1]. 1.2. Properties of the... |

42 |
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Citation Context ..., Fellowship BD 2366/92-RM and Program PRAXIS XXI, Fellowship BD 5236/95, Junta Nacional de Investigação Científica e Tecnológica, Portugal, and by a Fulbright/FLAD Grant. 1 Δ 2s2 LUIS M. BARREIR=-=A In [Mo], Moran -=-showed that the Hausdorff dimension of F is the unique root s of the equation p� k=1 λk s = 1. (2) We emphasize that this result does not depend on the location of the intervals ∆i1···in . The... |

40 |
Thermodynamic Formalism, Encyclopedia of Mathematics and
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Citation Context ...s is called a Markov partition of X (with respect to h) if: (a) int R i = R i for each i = 1, : : : , p; (b) int R i \ int R j = ? when i 6= j; (c) each hR i is a union of sets R j . Ruelle showed in =-=[R2]-=- that any repeller of a continuous expanding map has Markov partitions of arbitrarily small diameter. A more explicit construction was recently carried out in [PeW2]. Let fR 1 , : : : , R p g be a Mar... |

33 |
A variational principle for the pressure of continuous transformations (to appear
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Citation Context ...racteristic. The new pressure generalizes the well-known notion of topological pressure with respect to a Z-action on a compact set, introduced by Ruelle in [R1] for expansive maps, and by Walters in =-=[W]-=- in the general case. For arbitrary sets (not necessarily compact), it generalizes the notion of topological pressure introduced by Pesin and Pitskel' in [PP], and the notions of lower and upper capac... |

28 |
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Citation Context ...ov partition has suciently small diameter, we have qsp. In addition, for each positive integer n, the number of sets R j 1 j n which intersect asxed R i 1 i n is at most q. By the discussion in [R3], it follows that for each n 1, we can choosesn > 0 such that each ball of radiussn intersects at most q of the sets R i 1 i n . We may assume that jX j > 0. Otherwise dimH X = s k = 0. Fix a suc... |

26 |
Symbolic dynamics and hyperbolic dynamic systems
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Citation Context ... (y) consists of a single point, which we denote [x; y]. We require the continuity of the map [ ; ]: (x; y) 2 X X : d(x; y)s! X: This denition is equivalent to the original denition given in [AY], and is formally contained in former work of Bowen. One can check that f is an Axiom A ] homeomorphism if and only if X is a Smale space with respect to f and [ ; ] (see [R2]). The point x 2 X is... |

25 |
R.,Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete 8
- Mañé
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Citation Context ...t limn→∞ ϕn/n = � ψ µ-almost everywhere and in L 1 (X, µ). Proof of Lemma 1. We observe that ψ ∈ L 1 (X, µ) and limn→∞(ϕn+1 − ϕn ◦ f) = ψ µ-almost everywhere and in L 1 (X, �=-=�). By Corollary II.1.6 in [Ma] we have � 1 lim ϕn+1 −ϕ1 ◦f n→∞ n n �-=-�� n−1 � k=0 ψ ◦f k � n−1 1 � = lim (ϕn−k+1 −ϕn−k ◦f −ψ)◦f n→∞ n k = 0 µ-almost everywhere and in L1 (X, µ). By Birkhoff’s ergodic theorem, there exists an f-invari... |

23 |
Hausdorff dimension of quasicircles, Inst
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- 1979
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Citation Context ...equation P f (s ') = 0. Here P f is the classical topological pressure with respect to some homeomorphism f : X ! X , and ' is a continuous function on X . Bowen's equation was introduced by Bowen in =-=[Bo3]-=-. This equation establishes the connection between the thermodynamic formalism and dimension theory: its unique root often gives the exact value or a good estimate of dimension. A good way to think ab... |

23 |
Dimensions and measures of quasi self-similar sets
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Citation Context ... the formula giving the Hausdor dimension of repellers of conformal expanding maps, established by Ruelle in [R3], and of the coincidence of their Hausdor and box dimensions obtained by Falconer in [F=-=2]-=-. In addition, we establish thesrst general dimension estimates for repellers of C 1 expanding maps that may not be conformal. Our methods of proof were inspired in the proofs of Palis and Takens in [... |

23 |
Symbolic dynamics and transformations of the unit interval
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- 1966
(Show Context)
Citation Context ...ralized Moran constructions, with a transitive transfer matrix and an expanding induced map, the roots are the same. Let F be the limit set of a general symbolic construction. By a result of Parry in =-=[Pa-=-], the induced map g: F ! F is a local homeomorphism at every point if and only if the sub-shift : Q ! Q is topologically conjugate to a Markov chain. Hence, the map g may not be expanding. However, a... |

22 | On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture
- Pesin, Weiss
- 1996
(Show Context)
Citation Context ...= Q n k=1 i k for somesxed numbers 0s1 , : : : , ps1. A NON-ADDITIVE THERMODYNAMIC FORMALISM AND DIMENSION THEORY 21 These constructions were introduced in [Mo] in the case of the full shift. See [P=-=-=-eW1] for the discussion of the case of general symbolic dynamics for a much broader class of constructions. We notice that the diameters of the basic sets are of a very special form: the ratio j i 1 i... |

21 |
Hausdorff dimension for horseshoes, Ergodic Theory Dynam
- McCluskey, Manning
- 1983
(Show Context)
Citation Context ...e case of invertible maps, we recover the formula giving the Hausdor di4 LUIS M. BARREIRA mension of basic sets of saddle-type Axiom A surface dieomorphisms, established by McCluskey and Manning in [MM]. Moreover, we obtain a new derivation of the coincidence of their Hausdor and box dimensions, obtained by Takens in [T] for C 2 dieomorphisms, and by Palis and Viana in [PV] for arbitrary C 1 dieo... |

19 |
Topological pressure and the variational principle for noncompact sets
- Pesin, Pitskel
- 1984
(Show Context)
Citation Context ...[R1] for expansive maps, and by Walters in [W] in the general case. For arbitrary sets (not necessarily compact), it generalizes the notion of topological pressure introduced by Pesin and Pitskel' in =-=[PP]-=-, and the notions of lower and upper capacity topological pressures introduced by Pesin in [Pe1]. The non-additive thermodynamic formalism contains as a particular case a new formulation of the sub-ad... |

18 |
Statistical mechanics on a compact set with Z v action satisfying expansiveness and specification
- Ruelle
- 1973
(Show Context)
Citation Context ...e pressure as a Caratheodory dimension characteristic. The new pressure generalizes the well-known notion of topological pressure with respect to a Z-action on a compact set, introduced by Ruelle in [=-=R1]-=- for expansive maps, and by Walters in [W] in the general case. For arbitrary sets (not necessarily compact), it generalizes the notion of topological pressure introduced by Pesin and Pitskel' in [PP]... |

15 |
Bounded distortion and dimension for non-conformal repellers
- Falconer
(Show Context)
Citation Context ...nuous (see [R2, p.135]); in our case, if this property holds one obtains dimension estimates using the numbers s and s for repellers of maps that may not be C 1+ or have -bunched derivative. In [F3]=-=-=-, Falconer considered C 2 expanding maps with 1-bunched derivative and obtained an upper estimate for dimB J that is in general sharper than the one in Corollary 3.11. Nevertheless, there are examples... |

14 |
Statistical mechanics on a compact set with Z ν action satisfying expansiveness and specification
- Ruelle
- 1973
(Show Context)
Citation Context ...ne pressure as a Carathéodory dimension characteristic. The new pressure generalizes the well-known notion of topological pressure with respect to a Z-action on a compact set, introduced by Ruelle in=-= [R1] f-=-or expansive maps, and by Walters in [W] in the general case. For arbitrary sets (not necessarily compact), it generalizes the notion of topological pressure introduced by Pesin and Pitskel’ in [PP]... |

13 |
Invariant measures of full dimension for some expanding maps. Ergodic Theory Dynam. Systems 17
- Gatzouras, Peres
- 1997
(Show Context)
Citation Context ...In [F2], Falconer showed that the Hausdor and box dimensions of such repellers coincide. For C 1 conformal expanding maps, the statement of Theorem 3.6 was also established by Gatzouras and Peres in [=-=GP-=-], and by Takens in [T] for dynamically dened Cantor sets. The following example demonstrates that the converse of Proposition 3.5 does not hold. Example 3.7. There is a C 1 expanding map g: X ! X on ... |

13 |
On the continuity of the Hausdorff dimension and limit capacity for horseshoes
- Palis, Viana
- 1988
(Show Context)
Citation Context ...luskey and Manning in [MM]. Moreover, we obtain a new derivation of the coincidence of their Hausdorff and box dimensions, obtained by Takens in [T] for C 2 diffeomorphisms, and by Palis and Viana in =-=[PV]-=- for arbitrary C 1 diffeomorphisms. In addition, we consider the case where both the dynamics along the stable and unstable foliations may not be conformal. Our approach is entirely based on the therm... |

10 |
Limit capacity and Hausdorff dimension of dynamically defined Cantor sets
- Takens
- 1988
(Show Context)
Citation Context ...ype Axiom A surface diffeomorphisms, established by McCluskey and Manning in [MM]. Moreover, we obtain a new derivation of the coincidence of their Hausdorff and box dimensions, obtained by Takens in =-=[T]-=- for C 2 diffeomorphisms, and by Palis and Viana in [PV] for arbitrary C 1 diffeomorphisms. In addition, we consider the case where both the dynamics along the stable and unstable foliations may not b... |

9 |
A subadditive thermodynamic formalism for mixing repellers
- Falconer
- 1988
(Show Context)
Citation Context ... pressures introduced by Pesin in [Pe1]. The non-additive thermodynamic formalism contains as a particular case a new formulation of the sub-additive thermodynamic formalism introduced by Falconer in =-=[F1]-=-. We establish a non-additive version of the classical variational principle. In particular, the new principle generalizes the variational principle obtained by Pesin and Pitskel' in [PP] in the addit... |

9 |
Stable manifolds and hyperbolic sets, Global Analysis (Proc
- Hirsch, Pugh
- 1968
(Show Context)
Citation Context ...ems 3.13 and 3.15. These are respectively slight modications of the proofs of Theorems 3.2 and 3.4. Proof of Theorem 3.20. It is well known that foliations of codimension one are of class C 1 (see [HP]). Hence, one can parameterize [W u (x; ); W s (x; )] by W u (x; ) W s (x; ) using a Lipschitz map with Lipschitz inverse. This implies that dimH [W u (x; ); W s (x; )] = dimH W u (x; ) W s ... |

8 |
Additive functions of intervals and Hausdor# measure
- Moran
- 1946
(Show Context)
Citation Context ...A, Fellowship BD 2366/92-RM and Program PRAXIS XXI, Fellowship BD 5236/95, Junta Nacional de Investigac~ao Cientca e Tecnologica, Portugal, and by a Fulbright/FLAD Grant. 1 2 LUIS M. BARREIRA In [Mo], Moran showed that the Hausdor dimension of F is the unique root s of the equation p X k=1 k s = 1: (2) We emphasize that this result does not depend on the location of the intervals i 1 i n ... |

8 | A multifractal analysis of Gibbs measures for conformal expanding maps and Markov Moran geometric constructions
- Pesin, Weiss
- 1997
(Show Context)
Citation Context ...union of sets R j . Ruelle showed in [R2] that any repeller of a continuous expanding map has Markov partitions of arbitrarily small diameter. A more explicit construction was recently carried out in =-=[PeW-=-2]. Let fR 1 , : : : , R p g be a Markov partition of X , with suciently small diameter so that no set R i intersects all the others simultaneously. We dene a pp transfer matrix A = (a ij ) with a ij ... |

7 |
Dimension type characteristics for invariant sets of dynamical systems
- Pesin
(Show Context)
Citation Context ...We develop a non-additive version of the topological pressure, for arbitrary sequences of functions ' n that may neither satisfy additivity nor even sub-additivity. We follow the approach of Pesin in =-=[Pe-=-1] to dene pressure as a Caratheodory dimension characteristic. The new pressure generalizes the well-known notion of topological pressure with respect to a Z-action on a compact set, introduced by Ru... |

7 | The dimensions of some self-affine limit sets in the plane and hyperbolic
- Pollicott, Weiss
- 1994
(Show Context)
Citation Context ...estimate in Corollary 3.11 can not be improved. We note there are non-conformal repellers for which the Hausdorff and box dimensions do not coincide. An example is described by Pollicott and Weiss in =-=[PoW],-=- modifying a construction of Przytycki and Urbański in [PU] that depends on delicate number-theoretic properties of the Lyapunov exponents. The following example illustrates that when g is not confor... |

6 |
Hölder continuity of the holonomy maps for hyperbolic sets
- Schmeling
- 1994
(Show Context)
Citation Context ...x2 r u (x) + r s (x) : For hyperbolic sets of C 2 dieomorphisms, an eective estimate of the largest Holder exponent of the holonomy maps was established by Schmeling and Siegmund -Schultze in [SS]. We now consider the case of surface dieomorphisms. We dene the functions ' u and ' s on by ' u (x) = log kd x f jE u k and ' s (x) = log kd x f 1 jE s k: Let t u and t s be the unique roots of t... |

5 |
Limit capacity and Hausdor dimension of dynamically de Cantor sets, Dynamical Systems (Valparaiso
- Takens
- 1986
(Show Context)
Citation Context ...-type Axiom A surface dieomorphisms, established by McCluskey and Manning in [MM]. Moreover, we obtain a new derivation of the coincidence of their Hausdor and box dimensions, obtained by Takens in [T] for C 2 dieomorphisms, and by Palis and Viana in [PV] for arbitrary C 1 dieomorphisms. In addition, we consider the case where both the dynamics along the stable and unstable foliations may not be ... |

5 |
On Hausdorff dimension of some fractal sets, Studia Mathematica 93
- Przytycki, Urba'nski
- 1989
(Show Context)
Citation Context ... are non-conformal repellers for which the Hausdorff and box dimensions do not coincide. An example is described by Pollicott and Weiss in [PoW], modifying a construction of Przytycki and Urbański in=-= [PU] t-=-hat depends on delicate number-theoretic properties of the Lyapunov exponents. The following example illustrates that when g is not conformal, the numbers s and s, and hence also the numbers s ∗ and... |

3 |
Ergodic theory and dierentiable dynamics, Ergebnisse der Mathematic und ihrer Grenzgebiete
- Mane
- 1987
(Show Context)
Citation Context ...that lim n!1 ' n =n = es-almost everywhere and in L 1 (X; ). Proof of Lemma 1. We observe thats2 L 1 (X; ) and lim n!1 (' n+1 ' n f) =s-almost everywhere and in L 1 (X; ). By Corollary II.1.6 in [Ma] we have lim n!1 1 n ' n+1 ' 1 f n n 1 X k=0sf k = lim n!1 1 n n 1 X k=0 (' n k+1 ' n k fs)f k = 0 -almost everywhere and in L 1 (X; ). By Birkho's ergodic theorem, there exists an f-invariant ... |

3 |
On the Hausdor dimension of some fractal sets Studia
- Przytycki, Urbanski
- 1989
(Show Context)
Citation Context ... are non-conformal repellers for which the Hausdor and box dimensions do not coincide. An example is described by Pollicott and Weiss in [PoW], modifying a construction of Przytycki and Urbanski in [P=-=U-=-] that depends on delicate number-theoretic properties of the Lyapunov exponents. The following example illustrates that when g is not conformal, the numbers s and s, and hence also the numbers s and... |

2 | Dimension of Cantor sets with complicated geometry
- Barreira
(Show Context)
Citation Context ...) Q-admissible r i 1 i n s = 0: (24) 20 LUIS M. BARREIRA The proof of Theorem 2.1 (see Section 4) shows that the inequalities in (a) still hold if the set Q is neither compact nor -invariant. See [Ba] for extensions to classes of limit sets with more complicated geometry. The following example demonstrates that we may have strict inequalities in Theorem 2.1a if the sequence is not sub-additive. ... |

2 |
Characteristic exponents of dynamical systems in metric spaces, Ergodic Theory Dynam. Systems 3
- Kifer
- 1983
(Show Context)
Citation Context ...If h is smooth, the numbers (!) and (!) respectively coincide with the inverse of the largest and smallest Lyapunov exponents of h at the point (!). Related notions weresrst considered by Kifer in [K]. 30 LUIS M. BARREIRA 3.1.2. Dimension estimates. We dene the sequences of functions ' k n and ' k n on + A by ' k n (!) = log k (!; n) and ' k n (!) = log k (!; n); and denote them k and k... |

2 |
On the continuity of Hausdor dimension and limit capacity for horseshoes
- Palis, Viana
- 1988
(Show Context)
Citation Context ...cCluskey and Manning in [MM]. Moreover, we obtain a new derivation of the coincidence of their Hausdor and box dimensions, obtained by Takens in [T] for C 2 dieomorphisms, and by Palis and Viana in [P=-=V-=-] for arbitrary C 1 dieomorphisms. In addition, we consider the case where both the dynamics along the stable and unstable foliations may not be conformal. Our approach is entirely based on the thermo... |

2 |
The dimensions of some self ane limit sets in the plane and hyperbolic
- Pollicott, Weiss
- 1994
(Show Context)
Citation Context ... estimate in Corollary 3.11 can not be improved. We note there are non-conformal repellers for which the Hausdor and box dimensions do not coincide. An example is described by Pollicott and Weiss in [=-=PoW-=-], modifying a construction of Przytycki and Urbanski in [PU] that depends on delicate number-theoretic properties of the Lyapunov exponents. The following example illustrates that when g is not confo... |