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February & March  2004, 11(2&3): 489-516. doi: 10.3934/dcds.2004.11.489

## Polynomial growth of the derivative for diffeomorphisms on tori

 1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Received  August 2002 Revised  February 2004 Published  June 2004

We consider area--preserving zero entropy ergodic diffeomorphisms on tori. We classify such diffeomorphisms for which the sequence {$Df^n$} has a polynomial growth on the $3$-torus: they are necessary of the form

$\mathbb T^3\quad (x_1,x_2,x_3)\mapsto (x_1+\alpha,\varepsilon x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\mathbb T^3, where$\varepsilon =\pm 1$. We also indicate why there is no$4$-dimensional analogue of the above result. Random diffeomorphisms on the$2$-torus are studied as well. Citation: Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489  [1] 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 [2] Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007 [3] Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231 [4] Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106 [5] Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123. [6] Masoumeh Gharaei, Ale Jan Homburg. Random interval diffeomorphisms. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 241-272. doi: 10.3934/dcdss.2017012 [7] Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611 [8] Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41 [9] Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555 [10] Lev Buhovski, Roman Muraviev. Growth gap versus smoothness for diffeomorphisms of the interval. Journal of Modern Dynamics, 2008, 2 (4) : 629-643. doi: 10.3934/jmd.2008.2.629 [11] Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61 [12] Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869 [13] João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837 [14] Radu Saghin. Volume growth and entropy for$C^1\$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789 [15] Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067 [16] Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581 [17] Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281 [18] Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615 [19] Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204 [20] Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075

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