6-  Lifting Measures to Inducing Schemes, with Yakov Pesin and Ke Zhang.   Ergod. Th. & Dynam. Sys., 28 (2), 553--574, 2008.

  Abstract: In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of [items 2 and 4 below]. We show that under some natural assumptions on the inducing schemes – which hold for many known examples – any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [Buz99], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [Bru95] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multimodal maps) and for some multidimensional maps.

 
5-  Affine Actions of a Free Semigroup on the Real Line, with Vitaly Bergelson and Michal Misiurewicz.   Ergod. Th. & Dynam. Sys. 26 (2006), 1285 - 1305   [dvi],   [pdf].

  Abstract: We consider actions of the free semigroup with two generators on the real line, where the generators act as affine maps, one contracting and one expanding, with distinct fixed points. Then every orbit is dense in a half-line, which leads to the question whether it is, in some sense, uniformly distributed. We present answers to this question for various interpretations of the phrase ``uniformly distributed''.

 
4-  Equilibrium Measures for Maps with Inducing Schemes,   with Yakov Pesin.   Journal of Modern Dynamics, Volume 2, No. 3, 1–31, 2008.

  Abstract: We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of real-valued potential functions $\varphi$ on I, which admit unique equilibrium measures $\mu$ minimizing the free energy for a certain class of measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the Central Limit Theorem. Our results apply in particular to some one-dimensional unimodal and multimodal maps as well as to multidimensional nonuniformly hyperbolic maps admitting Young’s tower. Examples of potential functions to which our theory applies include \varphi_t = −t log |df| with t_0

 
3-  Maximal Entropy Measures for Viana Maps,   with Alexander Arbieto and Carlos Matheus.   Preprint, 2005. Submitted.   [dvi],   [pdf].

  Abstract: In this note we construct measures of maximal entropy for a certain class of maps with critical points called Viana maps. The main ingredients of the proof are the non-uniform expansion features and the slow recurrence (to the critical set) of generic points with respect to the natural candidates for attaining the topological entropy.

 
2-  Thermodynamical Formalism Associated with Inducing Schemes for One Dimensional Maps,   with Yakov Pesin.   Moscow Mathematical Journal, 2005.   [dvi],   [pdf].

  Abstract: For a smooth map f of a compact interval I admitting an inducing scheme we establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions \varphi on I which admit a unique equilibrium measure \mu_\varphi. Our results apply to unimodal maps corresponding to a positive Lebesgue measure set of parameters in a one-parameter transverse family.

 
1-  Dimension of Weakly Expanding Points for Quadratic Maps.   Bull. Soc. Math. France, 131 (3) : 399-420, 2003.   [ps],   [pdf].   Abstract [pdf]

0-  Hausdorff Dimension of the Exceptional set in Jakobson's Theorem .ps, .pdf.
Ph.D. Thesis (Advisor: J.-C. Yoccoz)
Abstract (.pdf)

Algebraic Topology: