Symmetry breaking in a globally coupled map of four sites

Previous Article
Degenerate lower dimensional invariant tori in reversible system
DCDS Home
This Issue
Next Article

August 2018, 38(8): 3707-3734. doi: 10.3934/dcds.2018161

Symmetry breaking in a globally coupled map of four sites

Fanni M. Sélley ^1,2,,

1.	Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053 Budapest, Hungary
2.	MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Egry József u. 1, H-1111 Budapest, Hungary

Received April 2017 Revised March 2018 Published May 2018

A system of four globally coupled doubling maps is studied in this paper. It is known that such systems have a unique absolutely continuous invariant measure (acim) for weak interaction, but the case of stronger coupling is still unexplored. As in the case of three coupled sites [14], we prove the existence of a critical value of the coupling parameter at which multiple acims appear. Our proof has several new ingredients in comparison to the one presented in [14]. We strongly exploit the symmetries of the dynamics in the course of the argument. This simplifies the computations considerably, and gives us a precise description of the geometry and symmetry properties of the arising asymmetric invariant sets. Some new phenomena are observed which are not present in the case of three sites. In particular, the asymmetric invariant sets arise in areas of the phase space which are transient for weaker coupling and a nontrivial symmetric invariant set emerges, shaped by an underlying centrally symmetric Lorenz map. We state some conjectures on further invariant sets, indicating that unlike the case of three sites, ergodicity breaks down in many steps, and not all of them are accompanied by symmetry breaking.

Keywords: Coupled map systems, ergodicity breaking, asymmetric invariant set, piecewise affine maps, circle maps.

Mathematics Subject Classification: Primary: 37A25, 37E10; Secondary: 37G99.

Citation: Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161

References:

[1]	C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti, Ising-type transitions in coupled map lattices, Journal of Statistical Physics, 80 (1995), 1185-1205. doi: 10.1007/BF02179868. Google Scholar
[2]	C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti, Ising-type and other transitions in one-dimensional coupled map lattices with sign symmetry, Journal of Statistical Physics, 102 (2001), 1271-1283. doi: 10.1023/A:1004892312745. Google Scholar
[3]	L. A. Bunimovich and Y. G. Sinai, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1998), 491-516. doi: 10.1088/0951-7715/1/4/001. Google Scholar
[4]	J. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, volume 671, Springer Science & Business Media, 2005. Google Scholar
[5]	B. Fernandez, Breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps, Journal of Statistical Physics, 154 (2014), 999-1029. doi: 10.1007/s10955-013-0903-9. Google Scholar
[6]	G. Gielis and R. S. MacKay, Coupled map lattices with phase transition, Nonlinearity, 13 (2000), 867-888. doi: 10.1088/0951-7715/13/3/320. Google Scholar
[7]	E. Järvenpää, A SRB-measure for globally coupled circle maps, Nonlinearity, 6 (1997), 1435.Google Scholar
[8]	M. Jiang and Y.B. Pesin, Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Communications in Mathematical Physics, 193 (1998), 675-711. doi: 10.1007/s002200050344. Google Scholar
[9]	W. Just, Globally coupled maps: Phase transitions and synchronization, Physica D: Nonlinear Phenomena, 81 (1995), 317-340. doi: 10.1016/0167-2789(94)00213-A. Google Scholar
[10]	G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Communications in Mathematical Physics, 262 (2006), 33-50. doi: 10.1007/s00220-005-1474-7. Google Scholar
[11]	J. Koiller and L. S. Young, Coupled map networks, Nonlinearity, 23 (2010), 1121-1141. doi: 10.1088/0951-7715/23/5/006. Google Scholar
[12]	J. Miller and D. A. Huse, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice, Physical Review E, 48 (1993), 2528. doi: 10.1103/PhysRevE.48.2528. Google Scholar
[13]	W. Parry, The Lorenz attractor and a related population model, in Ergodic Theory, pages 169–187, Lecture Notes in Math., 729, Springer, Berlin, 1979. Google Scholar
[14]	F. Sélley and P. Bálint, Mean-field coupling of identical expanding circle maps, Journal of Statistical Physics, 164 (2016), 858-889. doi: 10.1007/s10955-016-1568-y. Google Scholar
[15]	D. Thomine, A spectral gap for transer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917–944, arXiv: 1006.2608. Google Scholar

show all references

References:

[1]	C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti, Ising-type transitions in coupled map lattices, Journal of Statistical Physics, 80 (1995), 1185-1205. doi: 10.1007/BF02179868. Google Scholar
[2]	C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti, Ising-type and other transitions in one-dimensional coupled map lattices with sign symmetry, Journal of Statistical Physics, 102 (2001), 1271-1283. doi: 10.1023/A:1004892312745. Google Scholar
[3]	L. A. Bunimovich and Y. G. Sinai, Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1998), 491-516. doi: 10.1088/0951-7715/1/4/001. Google Scholar
[4]	J. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, volume 671, Springer Science & Business Media, 2005. Google Scholar
[5]	B. Fernandez, Breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps, Journal of Statistical Physics, 154 (2014), 999-1029. doi: 10.1007/s10955-013-0903-9. Google Scholar
[6]	G. Gielis and R. S. MacKay, Coupled map lattices with phase transition, Nonlinearity, 13 (2000), 867-888. doi: 10.1088/0951-7715/13/3/320. Google Scholar
[7]	E. Järvenpää, A SRB-measure for globally coupled circle maps, Nonlinearity, 6 (1997), 1435.Google Scholar
[8]	M. Jiang and Y.B. Pesin, Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Communications in Mathematical Physics, 193 (1998), 675-711. doi: 10.1007/s002200050344. Google Scholar
[9]	W. Just, Globally coupled maps: Phase transitions and synchronization, Physica D: Nonlinear Phenomena, 81 (1995), 317-340. doi: 10.1016/0167-2789(94)00213-A. Google Scholar
[10]	G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Communications in Mathematical Physics, 262 (2006), 33-50. doi: 10.1007/s00220-005-1474-7. Google Scholar
[11]	J. Koiller and L. S. Young, Coupled map networks, Nonlinearity, 23 (2010), 1121-1141. doi: 10.1088/0951-7715/23/5/006. Google Scholar
[12]	J. Miller and D. A. Huse, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice, Physical Review E, 48 (1993), 2528. doi: 10.1103/PhysRevE.48.2528. Google Scholar
[13]	W. Parry, The Lorenz attractor and a related population model, in Ergodic Theory, pages 169–187, Lecture Notes in Math., 729, Springer, Berlin, 1979. Google Scholar
[14]	F. Sélley and P. Bálint, Mean-field coupling of identical expanding circle maps, Journal of Statistical Physics, 164 (2016), 858-889. doi: 10.1007/s10955-016-1568-y. Google Scholar
[15]	D. Thomine, A spectral gap for transer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917–944, arXiv: 1006.2608. Google Scholar

Figure 1. The function $g$.

Download full-size image

Download as PowerPoint slide

Figure 2. The graph of $L_{\varepsilon}$.

Download full-size image

Download as PowerPoint slide

Figure 3. The asymmetric set $\mathcal{A}$.

Download full-size image

Download as PowerPoint slide

Figure 4. Six asymmetric invariant sets and the symmetries connecting them. Edge colors indicate the symmetries as according to the legend.

Download full-size image

Download as PowerPoint slide

Figure 5. Eight conjectured asymmetric invariant sets and the symmetries connecting them. Edge colors indicate the symmetries according to the legend.

Download full-size image

Download as PowerPoint slide

Figure 6. The symmetric set $\mathcal{S}$ for $1-\frac{\sqrt{2}}{2} \leq \varepsilon$.

Download full-size image

Download as PowerPoint slide

Figure 7. Overview of the critical parameters and the corresponding invariant sets. Results are marked with black, conjectures with gray.

Download full-size image

Download as PowerPoint slide

Figure 8. Continuity domains of the map $G_{\varepsilon,3}$. $\mathbb{T}^3$ is represented as the unit cube of $\mathbb{R}^3$. The singularities are marked according to the legend.

Download full-size image

Download as PowerPoint slide

Figure 9. Cubes $1$ and $2$.

Download full-size image

Download as PowerPoint slide

Figure 10. Cubes $3$ and $4$.

Download full-size image

Download as PowerPoint slide

Figure 11. Cubes $5$ and $6$.

Download full-size image

Download as PowerPoint slide

Figure 12. Cubes 7 and 8.

Download full-size image

Download as PowerPoint slide

Figure 13. The image of $P_1 \cap 4a$ for $\varepsilon = 0.37 < \varepsilon^*$ and $\varepsilon = 0.41 > \varepsilon^*$.

Download full-size image

Download as PowerPoint slide

Figure 14. The sets $\mathcal{A}$ (black) and $S_0(\mathcal{A})$ (gray). The plane $r-p = 1-2p^*$ separates $S_0(P_1)$, $S_0(P_2)$ and $\mathcal{A}$.

Download full-size image

Download as PowerPoint slide

Figure 15. The sets $\mathcal{A}$ (black) and $S_1(\mathcal{A})$ (gray). Different angles are plotted for better visibility.

Download full-size image

Download as PowerPoint slide

Figure 16. Images of $P_0 \cap 4b$ and $P_0 \cap 1e$ for $\varepsilon = 0.32 > 1-\frac{\sqrt{2}}{2}$.

Download full-size image

Download as PowerPoint slide

Table 1. Domains of continuity contained in Cube 1.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{1a}$	$p > 0$	$q > 0$	$r > 0$			$p+q+r < 1/2$
$\bf{1b}$	$p < 1/2$		$r > 0$	$p+q > 1/2$	$q+r < 1/2$
$\bf{1c}$	$p > 0$		$r < 1/2$	$p+q < 1/2$	$q+r > 1/2$
$\bf{1d}$		$q > 0 $		$p+q < 1/2$	$q+r < 1/2$	$p+q+r > 1/2$
$\bf{1e}$	$p < 1/2$	$q < 1/2 $	$r < 1/2$	$p+q > 1/2$	$q+r > 1/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{1a}$	$p > 0$	$q > 0$	$r > 0$			$p+q+r < 1/2$
$\bf{1b}$	$p < 1/2$		$r > 0$	$p+q > 1/2$	$q+r < 1/2$
$\bf{1c}$	$p > 0$		$r < 1/2$	$p+q < 1/2$	$q+r > 1/2$
$\bf{1d}$		$q > 0 $		$p+q < 1/2$	$q+r < 1/2$	$p+q+r > 1/2$
$\bf{1e}$	$p < 1/2$	$q < 1/2 $	$r < 1/2$	$p+q > 1/2$	$q+r > 1/2$

Table 2. Domains of continuity contained in Cube 2.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{2a}$	$0 < p < 1/2$	$1/2 < q < 1 $	$0 < r < 1/2$			$p+q+r < 3/2$
$\bf{2b}$	$p < 1/2$	$ q < 1 $	$ r < 1/2$			$p+q+r > 3/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{2a}$	$0 < p < 1/2$	$1/2 < q < 1 $	$0 < r < 1/2$			$p+q+r < 3/2$
$\bf{2b}$	$p < 1/2$	$ q < 1 $	$ r < 1/2$			$p+q+r > 3/2$

Table 3. Domains of continuity contained in Cube 3.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{3a}$	$p >1/2$	$ q > 1/2 $	$ r < 1/2$	$p+q < 3/2$		$p+q+r > 3/2$
$\bf{3b}$	$p > 1/2$	$ q > 1/2 $	$ r > 0$			$p+q+r < 3/2$
$\bf{3c}$	$p < 1$	$ q < 1 $	$ 0 < r < 1/2$	$p+q > 3/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{3a}$	$p >1/2$	$ q > 1/2 $	$ r < 1/2$	$p+q < 3/2$		$p+q+r > 3/2$
$\bf{3b}$	$p > 1/2$	$ q > 1/2 $	$ r > 0$			$p+q+r < 3/2$
$\bf{3c}$	$p < 1$	$ q < 1 $	$ 0 < r < 1/2$	$p+q > 3/2$

Table 4. Domains of continuity contained in Cube 4.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{4a}$	$1/2 < p < 1$	$ q > 0$	$ r > 0$		$q+r < 1/2$
$\bf{4b}$	$p > 1/2$	$ q < 1/2$	$ r < 1/2$		$q+r > 1/2$	$p+q+r < 3/2$
$\bf{4c}$	$p < 1$	$ q < 1/2$	$ r < 1/2$			$p+q+r > 3/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{4a}$	$1/2 < p < 1$	$ q > 0$	$ r > 0$		$q+r < 1/2$
$\bf{4b}$	$p > 1/2$	$ q < 1/2$	$ r < 1/2$		$q+r > 1/2$	$p+q+r < 3/2$
$\bf{4c}$	$p < 1$	$ q < 1/2$	$ r < 1/2$			$p+q+r > 3/2$

Table 5. Domains of continuity contained in Cube 5.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{5a}$	$p > 0$	$ q > 0$	$1/2 < r < 1$	$p+q < 1/2$
$\bf{5b}$	$p < 1/2$	$ q < 1/2$	$r > 1/2$	$p+q > 1/2$		$p+q+r < 3/2$
$\bf{5c}$	$p < 1/2$	$ q < 1/2$	$r < 1$			$p+q+r > 3/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{5a}$	$p > 0$	$ q > 0$	$1/2 < r < 1$	$p+q < 1/2$
$\bf{5b}$	$p < 1/2$	$ q < 1/2$	$r > 1/2$	$p+q > 1/2$		$p+q+r < 3/2$
$\bf{5c}$	$p < 1/2$	$ q < 1/2$	$r < 1$			$p+q+r > 3/2$

Table 6. Domains of continuity contained in Cube 6.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{6a}$	$p < 1/2$	$ q > 1/2$	$r > 1/2$		$q+r < 3/2$	$p+q+r > 3/2$
$\bf{6b}$	$p > 0$	$ q > 1/2$	$r > 1/2$			$p+q+r < 3/2$
$\bf{6c}$	$0 < p < 1/2$	$ q < 1$	$r < 1$		$q+r > 3/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{6a}$	$p < 1/2$	$ q > 1/2$	$r > 1/2$		$q+r < 3/2$	$p+q+r > 3/2$
$\bf{6b}$	$p > 0$	$ q > 1/2$	$r > 1/2$			$p+q+r < 3/2$
$\bf{6c}$	$0 < p < 1/2$	$ q < 1$	$r < 1$		$q+r > 3/2$

Table 7. Domains of continuity contained in Cubes 7 and 8.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{7a}$	$p < 1$	$ q < 1$	$r < 1$			$p+q+r > 5/2$
$\bf{7b}$	$p > 1/2$		$r < 1$	$p+q < 3/2$	$q+r > 3/2$
$\bf{7c}$	$p < 1$		$r > 1/2$	$p+q > 3/2$	$q+r < 3/2$
$\bf{7d}$		$q < 1$		$p+q > 3/2$	$q+r > 3/2$	$p+q+r < 5/2$
$\bf{7e}$	$p > 1/2$	$q >1/2$	$r > 1/2$	$p+q < 3/2$	$q+r < 3/2$

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{8a}$	$1/2 < p < 1$	$0 < q < 1$	$1/2 < r < 1$			$p+q+r > 3/2$
$\bf{8b}$	$p > 1/2$	$q >0$	$r > 1/2$			$p+q+r < 3/2$

Download as excel

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{7a}$	$p < 1$	$ q < 1$	$r < 1$			$p+q+r > 5/2$
$\bf{7b}$	$p > 1/2$		$r < 1$	$p+q < 3/2$	$q+r > 3/2$
$\bf{7c}$	$p < 1$		$r > 1/2$	$p+q > 3/2$	$q+r < 3/2$
$\bf{7d}$		$q < 1$		$p+q > 3/2$	$q+r > 3/2$	$p+q+r < 5/2$
$\bf{7e}$	$p > 1/2$	$q >1/2$	$r > 1/2$	$p+q < 3/2$	$q+r < 3/2$

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{8a}$	$1/2 < p < 1$	$0 < q < 1$	$1/2 < r < 1$			$p+q+r > 3/2$
$\bf{8b}$	$p > 1/2$	$q >0$	$r > 1/2$			$p+q+r < 3/2$

Table 8. Values of $c_1,c_2$ and $c_3$ in Formula 5, for each continuity domain.

		$\bf{1a}$	$\bf{1b}$	$\bf{1c}$	$\bf{1d}$	$\bf{1e}$		$\bf{2a}$	$\bf{2b}$		$\bf{3a}$	$\bf{3b}$	$\bf{3c}$		$\bf{4a}$	$\bf{4b}$	$\bf{4c}$		$\bf{5a}$	$\bf{5b}$	$\bf{5c}$	$\bf{6a}$	$\bf{6b}$		$\bf{6c}$	$\bf{7a}$		$\bf{7b}$	$\bf{7c}$	$\bf{7d}$		$\bf{7e}$	$\bf{8a}$	$\bf{8b}$

$\bf{c_1}$		0	2	0	1	1		0	1		4	3	2		4	3	4		0	1	2	0	1		0	4		2	4	3		3	4	3
$\bf{c_2}$		0	1	1	0	2		4	4		4	3	3		0	1	1		0	1	1	4	3		3	4		3	3	4		2	0	0
$\bf{c_3}$		0	0	2	1	1		0	1		0	1	0		0	1	2		4	3	4	4	3		2	4		4	2	3		3	4	3

Download as excel

		$\bf{1a}$	$\bf{1b}$	$\bf{1c}$	$\bf{1d}$	$\bf{1e}$		$\bf{2a}$	$\bf{2b}$		$\bf{3a}$	$\bf{3b}$	$\bf{3c}$		$\bf{4a}$	$\bf{4b}$	$\bf{4c}$		$\bf{5a}$	$\bf{5b}$	$\bf{5c}$	$\bf{6a}$	$\bf{6b}$		$\bf{6c}$	$\bf{7a}$		$\bf{7b}$	$\bf{7c}$	$\bf{7d}$		$\bf{7e}$	$\bf{8a}$	$\bf{8b}$

$\bf{c_1}$		0	2	0	1	1		0	1		4	3	2		4	3	4		0	1	2	0	1		0	4		2	4	3		3	4	3
$\bf{c_2}$		0	1	1	0	2		4	4		4	3	3		0	1	1		0	1	1	4	3		3	4		3	3	4		2	0	0
$\bf{c_3}$		0	0	2	1	1		0	1		0	1	0		0	1	2		4	3	4	4	3		2	4		4	2	3		3	4	3

[1]	Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753
[2]	Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739
[3]	Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737
[4]	Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597
[5]	Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099
[6]	Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[7]	Peter Hinow, Ami Radunskaya. Ergodicity and loss of capacity for a random family of concave maps. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2193-2210. doi: 10.3934/dcdsb.2016043
[8]	Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91
[9]	Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533
[10]	Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[11]	Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685
[12]	Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451
[13]	Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[14]	Laura Poggiolini, Marco Spadini. Local inversion of a class of piecewise regular maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2207-2224. doi: 10.3934/cpaa.2018105
[15]	Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039
[16]	Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[17]	Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[18]	Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[19]	Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683
[20]	M. R. S. Kulenović, Orlando Merino. A global attractivity result for maps with invariant boxes. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 97-110. doi: 10.3934/dcdsb.2006.6.97

2018 Impact Factor: 1.143

Tools

Metrics

Other articles
by authors

on AIMS
Fanni M. Sélley
on Google Scholar
Fanni M. Sélley

2018 Impact Factor: 1.143

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.143

Tools

Figures and Tables

2018 Impact Factor: 1.143

Tools

Metrics

Other articles
by authors

on AIMS
Fanni M. Sélley
on Google Scholar
Fanni M. Sélley

[Back to Top]