zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On codings and dynamics of planar piecewise rotations. (English) Zbl 1246.37064
Summary: We investigate codings and dynamics of piecewise rotations belonging to a sub-class of piecewise isometries. We show that the irrational set is empty for some piecewise rational rotations under some assumptions, while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space. We further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation.
37E30Homeomorphisms and diffeomorphisms of planes and surfaces
37E45Rotation numbers and vectors
37B10Symbolic dynamics
[1]P. Ashwin, Non-smooth invariant circles in digital overflow oscillations, in: Proc. of the 4th Int. Workshop on Nonl. Dynam. of Electr. Syst., Sevilla, 1996, pp. 417–422.
[2]P. Ashwin, J.H.B. Deane, X.-C. Fu, Dynamics of a bandpass sigma–delta modulator as a piecewise isometry, in: Proceedings of ISCA, 2001.
[3]Chua, L. O.; Lin, T.: Chaos in digital filters, IEEE trans. Circuits syst. I 35, 648-658 (1988)
[4]O. Feely, Nonlinear dynamics of bandpass sigma–delta modulation, in: Proc. of NDES, Dublin, 1995, pp. 33–36.
[5]Haller, H.: Rectangle exchange transformations, Monatsh. math. 91, 215-232 (1981) · Zbl 0448.47004 · doi:10.1007/BF01301789
[6]Keane, M.: Interval exchange tranformations, Math. Z. 141, 25-31 (1975) · Zbl 0278.28010 · doi:10.1007/BF01236981
[7]Kocarev, L.; Wu, C. W.; Chua, L. O.: Complex behavior in digital filters with overflow nonlinearity: analytical results, IEEE trans. Circuits syst. II 43, 234-246 (1996)
[8]Scott, A. J.; Holmes, C. A.; Milburn, G. J.: Hamiltonian mappings and circle packing phase spaces, Physica D 155, 34-50 (2001) · Zbl 0981.37017 · doi:10.1016/S0167-2789(01)00263-9
[9]Veech, W. A.: Interval exchange transformations, J. anal. Math. 33, 222-272 (1978) · Zbl 0455.28006 · doi:10.1007/BF02790174
[10]Veech, W. A.: Gauss measures for transformations on the space of interval exchange maps, Ann. of math. 115, 201-242 (1982) · Zbl 0486.28014 · doi:10.2307/1971391
[11]Vivaldi, F.; Shaidenko, A. V.: Global stability of a class of discontinuous dual billiards, Comm. math. Phys. 110, 625-640 (1987) · Zbl 0653.58018 · doi:10.1007/BF01205552
[12]Ashwin, P.; Fu, X. -C.: On the geometry of orientation perserving planar piecewise isometries, J. nonlinear sci. 12, 207-240 (2002) · Zbl 1100.37027 · doi:10.1007/s00332-002-0477-1
[13]Buzzi, J.: Piecewise isometries have zero topological entropy, Ergodic theory dynam. Systems 21, 1371-1377 (2001) · Zbl 0993.37012 · doi:10.1017/S0143385701001651
[14]Deane, J. H. B.: Global attraction in the sigma–delta modulator piecewise isometry, Dyn. syst. 17, 377-388 (2002) · Zbl 1022.37026 · doi:10.1080/1468936021000043922
[15]Fu, X. -C.; Ashwin, P.: Symbolic analysis for some planar piecewise linear maps, Discrete contin. Dyn. syst. 9, 1533-1548 (2003) · Zbl 1042.37009 · doi:10.3934/dcds.2003.9.1533
[16]Goetz, A.: Dynamics of piecewise rotation, Discrete contin. Dyn. syst. 4, 593-608 (1998) · Zbl 0965.37037 · doi:10.3934/dcds.1998.4.593
[17]Mendes, M.; Nicol, M.: Periodicity and recurrence in piecewise rotations of Euclidean spaces, Internat. J. Bifur. chaos 14, 2353-2361 (2004) · Zbl 1077.37504 · doi:10.1142/S0218127404010813
[18]Yu, R. -Z.; Fu, X. -C.; Wang, K. -M.; Chen, Z. -H.: Dynamical behaviors and codings of invertible planar piecewise isometric systems, Nonlinear anal. 72, 3575-3582 (2010) · Zbl 1188.37046 · doi:10.1016/j.na.2009.12.036
[19]Goetz, A.: Dynamics of piecewise isometries, Illinois J. Math. 44, 465-478 (2000) · Zbl 0964.37009
[20]Goetz, A.: Stability of piecewise rotations and affine maps, Nonlinearity 14, 205-219 (2001) · Zbl 1031.37039 · doi:10.1088/0951-7715/14/2/302
[21]Mends, M.: Stability of periodic points in piecewise isometries of Euclidean spance, Ergodic theory dynam. Systems 27, 183-197 (2007) · Zbl 1131.37027 · doi:10.1017/S0143385706000460
[22]Kahng, B.: Redefining chaos: devaney-chaos for piecewise continuous dynamical systems, Internat. J. Math. model. Methods appl. Sci. 3, 317-326 (2009)
[23]Kanhng, B.: On denvaney’s definiton of chaos for discontinuous dynamical systems, Recent adv. Appl. math. Comput. info. Sci. 15, 89-94 (2009)
[24]Kanhng, B.: Singularities of two-dimensional invertible piecewise isometric dynamics, Chaos 19, 023115 (2009)
[25]Ashwin, P.; Fu, X. -C.: Tangencies in invariant disk packings for certain planar piecewise isometries are rare, Dyn. syst. 16, 333-345 (2001) · Zbl 1032.37026 · doi:10.1080/14689360110073650
[26]Trovati, M.; Ashwin, P.: Tangency properties of a pentagonal tiling generated by a piecewise isometry, Chaos 17, 043129 (2007) · Zbl 1163.37379 · doi:10.1063/1.2825291
[27]Trovati, M.; Ashwin, P.; Byott, N.: Packings induced by piecewise isometries cannot contain the arbelos, Discrete contin. Dyn. syst. Ser. A 22, 791-806 (2008) · Zbl 1156.52013 · doi:10.3934/dcds.2008.22.791 · doi:http://aimsciences.org/journals/pdfs.jsp?paperID=3559{&}mode=abstract
[28]Yu, R. -Z.; Fu, X. -C.; Shui, S. -L.: Density of invariant disks packing in planar piecewise isometries, Dyn. syst. 22, 65-72 (2007) · Zbl 1115.37045 · doi:10.1080/14689360601054759
[29]Mendes, M.: A note on the coding of orbits in certain discontinuous maps, Discrete contin. Dyn. syst. 27, 369-382 (2010) · Zbl 1196.37042 · doi:10.3934/dcds.2010.27.369
[30]Mendes, M.: Quasi-invariant attractors of piecewise isometric systems, Discrete contin. Dyn. syst. 9, 323-338 (2003) · Zbl 1025.37026 · doi:10.3934/dcds.2003.9.323
[31]Goetz, A.: Perturbations of 8-attractors and births of satellite systems, Internat. J. Bifur. chaos 8, 1937-1956 (1998) · Zbl 0956.37029 · doi:10.1142/S0218127498001613
[32]Fu, X. -C.; Shui, S. -L.; Yuan, L. -G.: Tangents-free property in the invariant disk packings generated by some general planar piecewise isometries, Chaos solitons fractals 36, 115-120 (2008) · Zbl 1152.52303 · doi:10.1016/j.chaos.2006.06.012