Removal of Phase Transition in the Chebyshev Quadratic and Thermodynamics for Hénon-Like Maps Near the First Bifurcation

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• 作者：Hiroki Takahasi
• 关键词：Chebyshev quadratic map ; Hénon ; like maps ; Thermodynamic formalism ; Phase transition
• 刊名：Journal of Statistical Physics
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：164
• 期：6
• 页码：1354-1378
• 全文大小：841 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613
• 卷排序：164

文摘

It is well-known that the geometric pressure function $$\begin{aligned} t\in {\mathbb {R}}\mapsto \sup _{\mu }\left\{ h_\mu (T_2)-t\int \log |dT_2(x)|d\mu (x)\right\} \end{aligned}$$of the Chebyshev quadratic map \(T_2(x)=1-2x^2\)\((x\in {\mathbb {R}})\) is not differentiable at \(t=-1\). We show that this phase transition can be “removed”, by an arbitrarily small singular perturbation of the map \(T_2\) into Hénon-like diffeomorphisms. A proof of this result relies on an elaboration of the well-known inducing techniques adapted to Hénon-like dynamics near the first bifurcation.KeywordsChebyshev quadratic mapHénon-like mapsThermodynamic formalismPhase transitionMathematics Subject Classification37D2537D3537G2582C26References1.Bedford, E., Smillie, J.: Real polynomial diffeomorphisms with maximal entropy: II. small Jacobian. Ergod. Theory Dyn. Syst. 26, 1259–1283 (2006)MathSciNetCrossRefMATHGoogle Scholar2.Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. 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Phys. 218, 1–97 (2001)ADSMathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer Science+Business Media New York 2016Authors and AffiliationsHiroki Takahasi1Email author1.Department of MathematicsKeio UniversityYokohamaJapan About this article CrossMark Print ISSN 0022-4715 Online ISSN 1572-9613 Publisher Name Springer US About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips