April  2011, 30(1): 313-363. doi: 10.3934/dcds.2011.30.313

Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

1. 

Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043

2. 

Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received  August 2009 Revised  October 2010 Published  February 2011

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.
Citation: Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313
References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory,", Mathematical Surveys and Monographs Vol. \textbf{50}, 50 (1997).   Google Scholar

[2]

R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $z^2+c_n$,, Pacific J. Math., 198 (2001), 347.  doi: 10.2140/pjm.2001.198.347.  Google Scholar

[3]

R. Brück, M. Büger and S. Reitz, Random iterations of polynomials of the form $z^2+c_n$: Connectedness of Julia sets,, Ergodic Theory Dynam. Systems, 19 (1999), 1221.  doi: 10.1017/S0143385799141658.  Google Scholar

[4]

M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289.  doi: 10.1017/S0143385797086458.  Google Scholar

[5]

M. Büger, On the composition of polynomials of the form $z^2+c_n$,, Math. Ann., 310 (1998), 661.   Google Scholar

[6]

L. Carleson, P. W. Jones and J. -C. Yoccoz, Julia and John,, Bol. Soc. Brazil. Math., 25 (1994), 1.   Google Scholar

[7]

M. Denker and M. Urbański, On the existence of conformal measures,, Trans. Amer. Math. Soc., 328 (1991), 563.  doi: 10.2307/2001795.  Google Scholar

[8]

M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps,, Nonlinearity, 4 (1991), 103.  doi: 10.1088/0951-7715/4/1/008.  Google Scholar

[9]

M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere,, Nonlinearity, 4 (1991), 365.  doi: 10.1088/0951-7715/4/2/008.  Google Scholar

[10]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition. Studies in Nonlinearity, (1989).   Google Scholar

[11]

K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons, (1997).   Google Scholar

[12]

H. Federer, "Geometric Measure Theory,", Springer, (1969).   Google Scholar

[13]

J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar

[14]

Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system,, Ergodic Theory Dynam. Systems, 23 (2003), 1807.  doi: 10.1017/S0143385703000129.  Google Scholar

[15]

A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I,, Proc. London Math. Soc. (3), 73 (1996), 358.  doi: 10.1112/plms/s3-73.2.358.  Google Scholar

[16]

A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups,, Math. Z., 222 (1996), 161.   Google Scholar

[17]

M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,, Ergodic Theory Dynam. Systems, 3 (1983), 351.   Google Scholar

[18]

M. Martens, The existence of σ-finite invariant measures, Applications to real one-dimensional dynamics,, Front for the Math., ().   Google Scholar

[19]

P. Mattila, "Geometry of Sets and Measures in Euclidean spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995).   Google Scholar

[20]

R. D. Mauldin, T. Szarek and M. Urbański, Graph directed Markov systems on Hilbert spaces,, Math. Proc. Cambridge Phil. Soc., 147 (2009), 455.  doi: 10.1017/S0305004109002448.  Google Scholar

[21]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[22]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511543050.  Google Scholar

[23]

J. Milnor, "Dynamics in One Complex Variable (Third Edition),", Annals of Mathematical Studies, (2006).   Google Scholar

[24]

V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry,, preprint 2008, (2008).   Google Scholar

[25]

W. Parry, "Entropy and Generators in Ergodic Theory,", Mathematics Lecture Note Series, (1969).   Google Scholar

[26]

F. Przytycki and M. Urbański, "Fractals in the Plane - The Ergodic Theory Methods,", to be published from Cambridge University Press, ().   Google Scholar

[27]

D. Ruelle, "Thermodynamic Formalism,", Encyclopedia of Math. and Appl., 5 (1978).   Google Scholar

[28]

R. Stankewitz, Completely invariant Julia sets of polynomial semigroups,, Proc. Amer. Math. Soc., 127 (1999), 2889.  doi: 10.1090/S0002-9939-99-04857-1.  Google Scholar

[29]

R. Stankewitz, Completely invariant sets of normality for rational semigroups,, Complex Variables Theory Appl., 40 (2000), 199.   Google Scholar

[30]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's,, Proc. Amer. Math. Soc., 128 (2000), 2569.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar

[31]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.   Google Scholar

[32]

R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, to appear in Trans. Amer. Math. Soc., ().   Google Scholar

[33]

D. Steinsaltz, Random logistic maps and Lyapunov exponents,, Indag. Mathem., 12 (2001), 557.   Google Scholar

[34]

H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717.   Google Scholar

[35]

H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups,, Kodai Mathematical Journal, 21 (1998), 10.  doi: 10.2996/kmj/1138043831.  Google Scholar

[36]

H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar

[37]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.   Google Scholar

[38]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390.   Google Scholar

[39]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893.  doi: 10.1017/S0143385705000532.  Google Scholar

[40]

H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Appl. Math. Comput., 187 (2007), 489.  doi: 10.1016/j.amc.2006.08.149.  Google Scholar

[41]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.   Google Scholar

[42]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets,, Discrete and Continuous Dynamical Systems Ser. A, 29 (2011), 1205.  doi: 10.3934/dcds.2011.29.1205.  Google Scholar

[43]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().   Google Scholar

[44]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups III: Classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles,, Ergodic Theory Dynam. Systems, 30 (2010), 1869.  doi: 10.1017/S0143385709000923.  Google Scholar

[45]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint 2007, (2007).   Google Scholar

[46]

H. Sumi, Interaction cohomology of forward or backward self-similar systems,, Adv. Math., 222 (2009), 729.  doi: 10.1016/j.aim.2009.04.007.  Google Scholar

[47]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc., 102 (2011), 50.  doi: 10.1112/plms/pdq013.  Google Scholar

[48]

H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics,, preprint 2010, (2010).   Google Scholar

[49]

H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps,, Monatsh. Math., 156 (2009), 371.  doi: 10.1007/s00605-008-0016-8.  Google Scholar

[50]

H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups,, Ergodic Theory Dynam. Systems, 30 (2010), 601.  doi: 10.1017/S0143385709000297.  Google Scholar

[51]

M. Urbański, Rational functions with no recurrent critical points,, Ergodic Theory Dynam. Systems, 14 (1994), 391.   Google Scholar

[52]

M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics,, Ergodic Theory Dynam. Systems, 17 (1997), 1449.  doi: 10.1017/S014338579708646X.  Google Scholar

[53]

P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982).   Google Scholar

[54]

W. Zhou and and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Sci. Bulletin, 37 (1992), 969.   Google Scholar

show all references

References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory,", Mathematical Surveys and Monographs Vol. \textbf{50}, 50 (1997).   Google Scholar

[2]

R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $z^2+c_n$,, Pacific J. Math., 198 (2001), 347.  doi: 10.2140/pjm.2001.198.347.  Google Scholar

[3]

R. Brück, M. Büger and S. Reitz, Random iterations of polynomials of the form $z^2+c_n$: Connectedness of Julia sets,, Ergodic Theory Dynam. Systems, 19 (1999), 1221.  doi: 10.1017/S0143385799141658.  Google Scholar

[4]

M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289.  doi: 10.1017/S0143385797086458.  Google Scholar

[5]

M. Büger, On the composition of polynomials of the form $z^2+c_n$,, Math. Ann., 310 (1998), 661.   Google Scholar

[6]

L. Carleson, P. W. Jones and J. -C. Yoccoz, Julia and John,, Bol. Soc. Brazil. Math., 25 (1994), 1.   Google Scholar

[7]

M. Denker and M. Urbański, On the existence of conformal measures,, Trans. Amer. Math. Soc., 328 (1991), 563.  doi: 10.2307/2001795.  Google Scholar

[8]

M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps,, Nonlinearity, 4 (1991), 103.  doi: 10.1088/0951-7715/4/1/008.  Google Scholar

[9]

M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere,, Nonlinearity, 4 (1991), 365.  doi: 10.1088/0951-7715/4/2/008.  Google Scholar

[10]

R. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition. Studies in Nonlinearity, (1989).   Google Scholar

[11]

K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons, (1997).   Google Scholar

[12]

H. Federer, "Geometric Measure Theory,", Springer, (1969).   Google Scholar

[13]

J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar

[14]

Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system,, Ergodic Theory Dynam. Systems, 23 (2003), 1807.  doi: 10.1017/S0143385703000129.  Google Scholar

[15]

A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I,, Proc. London Math. Soc. (3), 73 (1996), 358.  doi: 10.1112/plms/s3-73.2.358.  Google Scholar

[16]

A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups,, Math. Z., 222 (1996), 161.   Google Scholar

[17]

M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,, Ergodic Theory Dynam. Systems, 3 (1983), 351.   Google Scholar

[18]

M. Martens, The existence of σ-finite invariant measures, Applications to real one-dimensional dynamics,, Front for the Math., ().   Google Scholar

[19]

P. Mattila, "Geometry of Sets and Measures in Euclidean spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995).   Google Scholar

[20]

R. D. Mauldin, T. Szarek and M. Urbański, Graph directed Markov systems on Hilbert spaces,, Math. Proc. Cambridge Phil. Soc., 147 (2009), 455.  doi: 10.1017/S0305004109002448.  Google Scholar

[21]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[22]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511543050.  Google Scholar

[23]

J. Milnor, "Dynamics in One Complex Variable (Third Edition),", Annals of Mathematical Studies, (2006).   Google Scholar

[24]

V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry,, preprint 2008, (2008).   Google Scholar

[25]

W. Parry, "Entropy and Generators in Ergodic Theory,", Mathematics Lecture Note Series, (1969).   Google Scholar

[26]

F. Przytycki and M. Urbański, "Fractals in the Plane - The Ergodic Theory Methods,", to be published from Cambridge University Press, ().   Google Scholar

[27]

D. Ruelle, "Thermodynamic Formalism,", Encyclopedia of Math. and Appl., 5 (1978).   Google Scholar

[28]

R. Stankewitz, Completely invariant Julia sets of polynomial semigroups,, Proc. Amer. Math. Soc., 127 (1999), 2889.  doi: 10.1090/S0002-9939-99-04857-1.  Google Scholar

[29]

R. Stankewitz, Completely invariant sets of normality for rational semigroups,, Complex Variables Theory Appl., 40 (2000), 199.   Google Scholar

[30]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's,, Proc. Amer. Math. Soc., 128 (2000), 2569.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar

[31]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.   Google Scholar

[32]

R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, to appear in Trans. Amer. Math. Soc., ().   Google Scholar

[33]

D. Steinsaltz, Random logistic maps and Lyapunov exponents,, Indag. Mathem., 12 (2001), 557.   Google Scholar

[34]

H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717.   Google Scholar

[35]

H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups,, Kodai Mathematical Journal, 21 (1998), 10.  doi: 10.2996/kmj/1138043831.  Google Scholar

[36]

H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar

[37]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.   Google Scholar

[38]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390.   Google Scholar

[39]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893.  doi: 10.1017/S0143385705000532.  Google Scholar

[40]

H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Appl. Math. Comput., 187 (2007), 489.  doi: 10.1016/j.amc.2006.08.149.  Google Scholar

[41]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.   Google Scholar

[42]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets,, Discrete and Continuous Dynamical Systems Ser. A, 29 (2011), 1205.  doi: 10.3934/dcds.2011.29.1205.  Google Scholar

[43]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().   Google Scholar

[44]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups III: Classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles,, Ergodic Theory Dynam. Systems, 30 (2010), 1869.  doi: 10.1017/S0143385709000923.  Google Scholar

[45]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint 2007, (2007).   Google Scholar

[46]

H. Sumi, Interaction cohomology of forward or backward self-similar systems,, Adv. Math., 222 (2009), 729.  doi: 10.1016/j.aim.2009.04.007.  Google Scholar

[47]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc., 102 (2011), 50.  doi: 10.1112/plms/pdq013.  Google Scholar

[48]

H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics,, preprint 2010, (2010).   Google Scholar

[49]

H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps,, Monatsh. Math., 156 (2009), 371.  doi: 10.1007/s00605-008-0016-8.  Google Scholar

[50]

H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups,, Ergodic Theory Dynam. Systems, 30 (2010), 601.  doi: 10.1017/S0143385709000297.  Google Scholar

[51]

M. Urbański, Rational functions with no recurrent critical points,, Ergodic Theory Dynam. Systems, 14 (1994), 391.   Google Scholar

[52]

M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics,, Ergodic Theory Dynam. Systems, 17 (1997), 1449.  doi: 10.1017/S014338579708646X.  Google Scholar

[53]

P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982).   Google Scholar

[54]

W. Zhou and and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Sci. Bulletin, 37 (1992), 969.   Google Scholar

[1]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[2]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[3]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[4]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[5]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[6]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[7]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[8]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[9]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[10]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122

[11]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011

[12]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[13]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[14]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[15]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[16]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[17]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[18]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[19]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[20]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]