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A spectral gap for transfer operators of piecewise expanding maps
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References:
[1] |
V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ Foliations, in "Algebraic and Topological Dynamics" (eds. Sergiy Kolyada, Yuri Manin and Thomas Ward), Contemporary Mathematics (2005), 123-136. |
[2] |
V. Baladi, "Positive Transfer Operators and Decay Of Correlations," World scientific, 2000.
doi: 10.1142/9789812813633. |
[3] |
V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institut Henri Poincaré, Analyse non linéaire, 26 (2009), 1453-1481. |
[4] |
V. Baladi and S. Gouëzel, Banach spaces for piecewise cone hyperbolic maps, Journal of Modern Dynamics, 4 (2010), 91-137.
doi: 10.3934/jmd.2010.4.91. |
[5] |
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
doi: 10.1088/0951-7715/15/6/309. |
[6] |
J. Buzzi, Intrisic ergodicity of affine maps in $[0,1]^d$, Monatshefte für Mathematik, 124 (1997), 97-118. |
[7] |
J. Buzzi, No or infinitely many A.C.I.P. for piecewise expanding $C^r$ maps in higher dimensions, Communications in Mathematical Physics, 222 (2001), 495-501.
doi: 10.1007/s002200100509. |
[8] |
W. J. Cowieson, Stochastic stability for piecewise expanding maps in $\mathbb{R}^{d}$, Nonlinearity, 13 (2000), 1745-1760.
doi: 10.1088/0951-7715/13/5/316. |
[9] |
W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory and Dynamical Systems, 22 (2002), 1061-1078.
doi: 10.1017/S0143385702000627. |
[10] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhaüser, Boston, 1984. |
[11] |
P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $\mathbb{R}^{n}$, Israël Journal of Mathematics, 67 (1989), 272-290.
doi: 10.1007/BF02764946. |
[12] |
H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, (French) Proceedings of the American Mathematical Society, 118 (1993), 627-639. |
[13] |
G. Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan, (French) Comptes-rendus de l'Académie des Sciences de Paris, 289 (1979), 625-627. |
[14] |
G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Zeitschrift für Wahrscheinlichkeitheorie und verwandte Geliete, 69 (1985), 461-478. |
[15] |
G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730. |
[16] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society, 186 (1973), 481-488. |
[17] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israël Journal of Mathematics, 116 (2000), 223-248. |
[18] |
R. S. Strichartz, Multipliers on fractional Sobolev spaces, Journal of Mathematics and Mechanics, 16 (1967), 1031-1060. |
[19] |
H. Triebel, General function spaces. III. (Spaces $B_{p,q}^{g(x)}$ and $F_{p,q}^{g(x)}$, $1 : Basic properties), Analysis Mathematica, 3 (1977), 221-249. |
[20] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978. |
[21] |
H. Triebel, "Theory of Function Spaces. II," Birkhäuser, Basel, 1992. |
[22] |
M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties, Ergodic Theory and Dynamical Systems, 20 (2000), 1851-1857. |
show all references
References:
[1] |
V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $C^\infty$ Foliations, in "Algebraic and Topological Dynamics" (eds. Sergiy Kolyada, Yuri Manin and Thomas Ward), Contemporary Mathematics (2005), 123-136. |
[2] |
V. Baladi, "Positive Transfer Operators and Decay Of Correlations," World scientific, 2000.
doi: 10.1142/9789812813633. |
[3] |
V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institut Henri Poincaré, Analyse non linéaire, 26 (2009), 1453-1481. |
[4] |
V. Baladi and S. Gouëzel, Banach spaces for piecewise cone hyperbolic maps, Journal of Modern Dynamics, 4 (2010), 91-137.
doi: 10.3934/jmd.2010.4.91. |
[5] |
M. Blank, G. Keller and C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
doi: 10.1088/0951-7715/15/6/309. |
[6] |
J. Buzzi, Intrisic ergodicity of affine maps in $[0,1]^d$, Monatshefte für Mathematik, 124 (1997), 97-118. |
[7] |
J. Buzzi, No or infinitely many A.C.I.P. for piecewise expanding $C^r$ maps in higher dimensions, Communications in Mathematical Physics, 222 (2001), 495-501.
doi: 10.1007/s002200100509. |
[8] |
W. J. Cowieson, Stochastic stability for piecewise expanding maps in $\mathbb{R}^{d}$, Nonlinearity, 13 (2000), 1745-1760.
doi: 10.1088/0951-7715/13/5/316. |
[9] |
W. J. Cowieson, Absolutely continuous invariant measures for most piecewise smooth expanding maps, Ergodic Theory and Dynamical Systems, 22 (2002), 1061-1078.
doi: 10.1017/S0143385702000627. |
[10] |
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Birkhaüser, Boston, 1984. |
[11] |
P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $\mathbb{R}^{n}$, Israël Journal of Mathematics, 67 (1989), 272-290.
doi: 10.1007/BF02764946. |
[12] |
H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, (French) Proceedings of the American Mathematical Society, 118 (1993), 627-639. |
[13] |
G. Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan, (French) Comptes-rendus de l'Académie des Sciences de Paris, 289 (1979), 625-627. |
[14] |
G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Zeitschrift für Wahrscheinlichkeitheorie und verwandte Geliete, 69 (1985), 461-478. |
[15] |
G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730. |
[16] |
A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of the American Mathematical Society, 186 (1973), 481-488. |
[17] |
B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israël Journal of Mathematics, 116 (2000), 223-248. |
[18] |
R. S. Strichartz, Multipliers on fractional Sobolev spaces, Journal of Mathematics and Mechanics, 16 (1967), 1031-1060. |
[19] |
H. Triebel, General function spaces. III. (Spaces $B_{p,q}^{g(x)}$ and $F_{p,q}^{g(x)}$, $1 : Basic properties), Analysis Mathematica, 3 (1977), 221-249. |
[20] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978. |
[21] |
H. Triebel, "Theory of Function Spaces. II," Birkhäuser, Basel, 1992. |
[22] |
M. Tsujii, Piecewise expanding maps on the plane with singular ergodic properties, Ergodic Theory and Dynamical Systems, 20 (2000), 1851-1857. |
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