# American Institute of Mathematical Sciences

January  2005, 12(1): 115-136. doi: 10.3934/dcds.2005.12.115

## A one-parameter family of analytic Markov maps with an intermittency transition

 1 Dipartimento di Matematica, Università di Roma 'Tor Vergata', Via della Ricerca Scientifica, I-00133 Roma, Italy 2 Dipartimento di Matematica e Informatica, Università di Camerino, via Madonna delle Carceri, I-62032 Camerino, Italy

Received  July 2003 Revised  September 2004 Published  December 2004

In this paper we introduce and study a one-parameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the Perron-Frobenius operator of all these maps and study the transition between discrete and continuous spectrum when approaching the intermittent situation.
Citation: Manuela Giampieri, Stefano Isola. A one-parameter family of analytic Markov maps with an intermittency transition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 115-136. doi: 10.3934/dcds.2005.12.115
 [1] Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917 [2] Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1 [3] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131 [4] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593 [5] Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018 [6] L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274 [7] Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651 [8] Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995 [9] Roland Martin. On simple Igusa local zeta functions. Electronic Research Announcements, 1995, 1: 108-111. [10] Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639 [11] Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435 [12] Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 279-305. doi: 10.3934/dcds.2006.16.279 [13] Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139 [14] Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015 [15] Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72 [16] Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821 [17] Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521 [18] João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465 [19] David Karpuk, Anne-Maria Ernvall-Hytönen, Camilla Hollanti, Emanuele Viterbo. Probability estimates for fading and wiretap channels from ideal class zeta functions. Advances in Mathematics of Communications, 2015, 9 (4) : 391-413. doi: 10.3934/amc.2015.9.391 [20] Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004

2018 Impact Factor: 1.143