Publications & Preprints

Visualization of 4D spaces and virtual reality.

• • More to come...

  • Polyvision: 4D space manipulation through multiple projections. (with K. Matsumoto, N. Ogawa, H. Inou, S. Kaji and M. Hirose) 

    Emerging Technologies, SIGGRAPH Asia 2019, Brisbane, Australia (2019). See also a YouTube video.

Hénon maps, Polynomial diffeomorphisms of C^2.

• The homotopy kernel. (with J. Smillie)  New! 

   Work in progress (2024).

• Pseudo-monodromy and the Mandelbrot set. (with T. Richards)  New! 

   Preprint (2024).

• Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus. (with Y. Hironaka) New!

   To appear in Ergodic Theory & Dynam. Systems (2024). 

• Boundary of the horseshoe locus for the Hénon family. (with Z. Arai and H. Takahasi) 

   SIAM J. Appl. Dyn. Syst. 17, no. 3, pp. 2234-2248 (2018). 

• On parameter loci of the Hénon family. (with Z. Arai)

   Commun. Math. Phys. 361, no. 2, pp. 343-414 (2018).

• Dynamics of polynomial diffeomorphisms of C^2: Combinatorial and topological aspects.

   Arnold Math. J. 3, no. 1, pp. 119-173 (2017). A special issue dedicated to the XXV anniversary of the IMS at Stony Brook.

• Hyperbolic polynomial diffeomorphisms of C^2. III: Iterated monodromy groups. 

   Adv. Math. 255, no. 1, pp. 242-304 (2014).

• Homotopy shadowing. (with J. Smillie) 

   Amer. J. Math. 132, no. 4, pp. 987-1029 (2010).

• Hyperbolic polynomial diffeomorphisms of C^2. II: Hubbard trees. 

   Adv. Math. 220, no. 4, pp. 985-1022 (2009). Corrigendum: Adv. Math. 226, no. 4, pp. 3850-3855 (2011).

• Hyperbolic polynomial diffeomorphisms of C^2. I: A non-planar map. 

   Adv. Math. 218, no. 2, pp. 417-464 (2008).

Lozi maps, Piecewise affine/projective maps.

• On some conjectures concerning the entropy of Lozi maps. (with D. Sands)  New! 

   Work in progress (2024).

• Lap number entropy formula for piecewise affine and projective maps in several dimensions. (with D. Sands) 

   Nonlinearity 20, no. 12, 2755-2772 (2007). Erratum (printing mistakes by publisher): Nonlinearity 21, no. 6, pp. 1411 (2008).

• Monotonicity of the Lozi family near the tent-maps. (with D. Sands) 

   Commun. Math. Phys. 198, no. 2, pp. 397-406 (1998).

• Towards a kneading theory for Lozi mappings II: Monotonicity of the topological entropy and Hausdorff dimension of attractors. 

   Commun. Math. Phys. 190, no. 2, pp. 375-394 (1997).

• Towards a kneading theory for Lozi mappings I: A solution of the pruning front conjecture and the first tangency problem. 

   Nonlinearity 10, no. 3, pp. 731-747 (1997). 

Fractal geometry and symbolic dynamics.

• Critical polyratio and core entropy. (with J. Park)  New! 

       Work in progress (2024).

• On the Hausdorff dimension of the recurrent sets induced from endomorphisms of free groups. (with T. Oka)

   J. Fractal Geom. 9, no. 1, pp. 171-192 (2022).

• M_4 is regular-closed. (with Y. Himeki)

   Ergodic Theory & Dynam. Systems 40, no. 1, pp. 213-220 (2020).

• Note on a paper by Kawasaki and Sasa on Bernoulli coupled map lattices. 

   J. Phys. A: Math. Gen. 39, no. 45, 14043-14046 (2006).

Physics-related dynamics, quantum tunneling.

• Julia sets and chaotic tunneling II. (with A. Shudo and K. S. Ikeda) 

   J. Phys. A: Math. Theor. 42, article number 265102 (34pp), (2009). 

• Julia sets and chaotic tunneling I. (with A. Shudo and K. S. Ikeda) 

   J. Phys. A: Math. Theor. 42, article number 265101 (26pp), (2009).

• Chaos attracts tunneling trajectories: A universal mechanism of chaotic tunneling. (with A. Shudo and K. S. Ikeda) 

   Europhys. Lett. 81, no. 5, 50003 (2008).

• Julia set describes quantum tunneling in the presence of chaos. (with A. Shudo and K. S. Ikeda) 

   J. Phys. A: Math. Gen. 35, no. 17, L225-L231 (2002).

• Quantum tunneling phenomena and complex dynamics. 

   Unpublished manuscript (1999).

Survey articles, lists of open problems, etc.

• Conjectures, problems and questions on the parameter space of the Hénon/Lozi family. (with Z. Arai)  New! 

   Submitted (2024). This paper contains an original result by M. Asaoka in Appendix.

• Hénon maps: a list of open problems. (Julia Xénelkis de Hénon)   New! 

    To appear in Arnold Math. J. (2024).

• Problems on complex Hénon maps. (in Japanese)

   Proceedings of the workshop "Problems on complex geometry", RIMS Kokyuroku 2211 (2022).

• Visualization of four-dimensional spaces using virtual reality. (in Japanese)

Suugaku Tsuushin 27, no. 3, pp. 49-54 (2022).

• Review of a paper by Sarah Koch. 

   Mathematical Reviews (2011). The content includes some useful remarks by E. Bedford and R. Dujardin.

• Describing the Julia sets by group actions. (in Japanese) 

   Notes for a lecture in an annual meeting of the Mathematical Society of Japan (2008).

• Discrete integrable systems and discrete dynamical systems. (in Japanese) 

   Suuri Kagaku (2003).

• Dynamics of the complex Hénon maps and the first tangency problem (after Bedford and Smillie). 

   Notes for lectures at Ryukoku University (2002).

• Physical phenomena and complex dynamics. (in Japanese) 

   Suuri Kagaku (2002).

• On some results of Bedford and Smillie concerning the complex dynamics of C^2. (in Japanese) 

   Kyoto University RIMS Kokyuroku, No. 1042, pp. 193-220 (1998).

Books.

• Japanese translation of "An Introduction to Chaotic Dynamical Systems" by R. Devaney (with H. Kokubu, M. Kisaka and S. Nii). 

   Kyoritsu Publisher.