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Geometric Pressure for Multimodal Maps of the Interval
 
Feliks Przytycki Polish Academy of Sciences, Warszawa, Poland
Juan Rivera-Letelier University of Rochester, Rochester, NY
Geometric Pressure for Multimodal Maps of the Interval
Softcover ISBN:  978-1-4704-3567-7
Product Code:  MEMO/259/1246
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5247-6
Product Code:  MEMO/259/1246.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3567-7
eBook: ISBN:  978-1-4704-5247-6
Product Code:  MEMO/259/1246.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
Geometric Pressure for Multimodal Maps of the Interval
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Geometric Pressure for Multimodal Maps of the Interval
Feliks Przytycki Polish Academy of Sciences, Warszawa, Poland
Juan Rivera-Letelier University of Rochester, Rochester, NY
Softcover ISBN:  978-1-4704-3567-7
Product Code:  MEMO/259/1246
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5247-6
Product Code:  MEMO/259/1246.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3567-7
eBook ISBN:  978-1-4704-5247-6
Product Code:  MEMO/259/1246.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2592019; 81 pp
    MSC: Primary 37

    This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism.

    The authors work in a setting of generalized multimodal maps, that is, smooth maps \(f\) of a finite union of compact intervals \(\widehat I\) in \(\mathbb{R}\) into \(\mathbb{R}\) with non-flat critical points, such that on its maximal forward invariant set \(K\) the map \(f\) is topologically transitive and has positive topological entropy. They prove that several notions of non-uniform hyperbolicity of \(f|_K\) are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in \(K\) hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). They prove that several definitions of geometric pressure \(P(t)\), that is pressure for the map \(f|_K\) and the potential \(-t\log |f'|\), give the same value (including pressure on periodic orbits, “tree” pressure, variational pressures and conformal pressure). Finally they prove that, provided all periodic orbits in \(K\) are hyperbolic repelling, the function \(P(t)\) is real analytic for \(t\) between the “condensation” and “freezing” parameters and that for each such \(t\) there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction: The main results
    • 2. Preliminaries
    • 3. Non-uniformly hyperbolic interval maps
    • 4. Equivalence of the definitions of geometric pressure
    • 5. Pressure on periodic orbits
    • 6. Nice inducing schemes
    • 7. Analytic dependence of geometric pressure on temperature. Equilibria
    • 8. Proof of key lemma: Induced pressure
    • A. More on generalized multimodal maps
    • B. Uniqueness of equilibrium via inducing
    • C. Conformal pressures
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2592019; 81 pp
MSC: Primary 37

This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism.

The authors work in a setting of generalized multimodal maps, that is, smooth maps \(f\) of a finite union of compact intervals \(\widehat I\) in \(\mathbb{R}\) into \(\mathbb{R}\) with non-flat critical points, such that on its maximal forward invariant set \(K\) the map \(f\) is topologically transitive and has positive topological entropy. They prove that several notions of non-uniform hyperbolicity of \(f|_K\) are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in \(K\) hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). They prove that several definitions of geometric pressure \(P(t)\), that is pressure for the map \(f|_K\) and the potential \(-t\log |f'|\), give the same value (including pressure on periodic orbits, “tree” pressure, variational pressures and conformal pressure). Finally they prove that, provided all periodic orbits in \(K\) are hyperbolic repelling, the function \(P(t)\) is real analytic for \(t\) between the “condensation” and “freezing” parameters and that for each such \(t\) there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.

  • Chapters
  • 1. Introduction: The main results
  • 2. Preliminaries
  • 3. Non-uniformly hyperbolic interval maps
  • 4. Equivalence of the definitions of geometric pressure
  • 5. Pressure on periodic orbits
  • 6. Nice inducing schemes
  • 7. Analytic dependence of geometric pressure on temperature. Equilibria
  • 8. Proof of key lemma: Induced pressure
  • A. More on generalized multimodal maps
  • B. Uniqueness of equilibrium via inducing
  • C. Conformal pressures
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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