Geometric and Functional Analysis

, Volume 29, Issue 5, pp 1325–1368 | Cite as

Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

  • Yongluo Cao
  • Yakov PesinEmail author
  • Yun Zhao


Given a non-conformal repeller \(\Lambda \) of a \(C^{1+\gamma }\) map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok’s approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure.

Keywords and phrases

Expanding map Repeller Topological pressure Non-uniform hyperbolicity theory Dimension 

Mathematics Subject Classification

37C45 37D35 37H15 



The authors would like to thank the referees for their careful reading and valuable comments which helped improve the paper substantially. The authors would like to thank Professor Dejun Feng and Wen Huang for their suggestions and comments. Ya. P. wants to thank Mittag-Leffler Institute and the Department of Mathematics of Weizmann Institute of Science where part of this work was done for their hospitality.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsSoochow UniversitySuzhouPeople’s Republic of China
  3. 3.Center for Dynamical Systems and Differential EquationSoochow UniversitySuzhouPeople’s Republic of China
  4. 4.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  5. 5.School of Mathematical SciencesSoochow UniversitySuzhouPeople’s Republic of China

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