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The following is not a solution but rather a reformulation of the original problem.

To make $f\colon B\to B$ invertible let us pass from the ball $B$ to the solenoid $S_f$, $$ S_f=\{\{x_n; n\in\mathbb Z\}: x_{n+1}=f(x_n)\} $$

Map $f$ induces the shift $f_*\colon S_f\to S_f$ and g induces $g_*(\{x_n\})=\{g(x_n)\}$. They commute and $f_*$ is invertible. If Brower's fixed point theorem were true for $S_f$ the result would follow. Indeed, a fixed point of $(f_*)^{-1}\circ g_*$ gives the desired orbit.

Solenoid $S_f$ is a non-empty closed subset of $B^{\mathbb Z}$ eqiupped with the product topology. Hence $S_f$ is compact. In fact, $S_f\subset K^{\mathbb Z}$, where $K=\cap_{n>0}f^n(B)$. And I think it is plausible (can anybody give a proof?) that $S_f$ contracts to the orbit of the fixed point of $f$. (later: I am not that sure now)

Unfortunately, Brower's fixed point theorem doesn't hold for compact contractible spaces. But this seem to be quite old and active research area as I learned from the introduction to this paper http://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02071-1/S0002-9947-99-02071-1.pdf People prove positive results under various additional assumptions.

So, maybe it's useful to look at the problem from this point of view.

The following is not a solution but rather a reformulation of the original problem.

To make $f\colon B\to B$ invertible let us pass from the ball $B$ to the solenoid $S_f$, $$ S_f=\{\{x_n; n\in\mathbb Z\}: x_{n+1}=f(x_n)\} $$

Map $f$ induces the shift $f_*\colon S_f\to S_f$ and g induces $g_*(\{x_n\})=\{g(x_n)\}$. They commute and $f_*$ is invertible. If Brower's fixed point theorem were true for $S_f$ the result would follow. Indeed, a fixed point of $f_*^{-1}\circ g_*$ gives the desired orbit.

Solenoid $S_f$ is a non-empty closed subset of $B^{\mathbb Z}$ eqiupped with the product topology. Hence $S_f$ is compact. In fact, $S_f\subset K^{\mathbb Z}$, where $K=\cap_{n>0}f^n(B)$. And I think it is plausible (can anybody give a proof?) that $S_f$ contracts to the orbit of the fixed point of $f$. (later: I am not that sure now)

Unfortunately, Brower's fixed point theorem doesn't hold for compact contractible spaces. But this seem to be quite old and active research area as I learned from the introduction to this paper http://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02071-1/S0002-9947-99-02071-1.pdf People prove positive results under various additional assumptions.

So, maybe it's useful to look at the problem from this point of view.

The following is not a solution but rather a reformulation of the original problem.

To make $f\colon B\to B$ invertible let us pass from the ball $B$ to the solenoid $S_f$, $$ S_f=\{\{x_n; n\in\mathbb Z\}: x_{n+1}=f(x_n)\} $$

Map $f$ induces the shift $f_*\colon S_f\to S_f$ and g induces $g_*(\{x_n\})=\{g(x_n)\}$. They commute and $f_*$ is invertible. If Brower's fixed point theorem were true for $S_f$ the result would follow. Indeed, a fixed point of ${f_{*}}^{-1}\circ g_*$ gives the desired orbit.

Solenoid $S_f$ is a non-empty closed subset of $B^{\mathbb Z}$ eqiupped with the product topology. Hence $S_f$ is compact. In fact, $S_f\subset K^{\mathbb Z}$, where $K=\cap_{n>0}f^n(B)$. And I think it is plausible (can anybody give a proof?) that $S_f$ contracts to the orbit of the fixed point of $f$.

Unfortunately, Brower's fixed point theorem doesn't hold for compact contractible spaces. But this seem to be quite old and active research area as I learned from the introduction to this paper http://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02071-1/S0002-9947-99-02071-1.pdf People prove positive results under various additional assumptions.

So, maybe it's useful to at the problem from this point of view.

The following is not a solution but rather a reformulation of the original problem.

To make $f\colon B\to B$ invertible let us pass from the ball $B$ to the solenoid $S_f$, $$ S_f=\{\{x_n; n\in\mathbb Z\}: x_{n+1}=f(x_n)\} $$

Map $f$ induces the shift $f_*\colon S_f\to S_f$ and g induces $g_*(\{x_n\})=\{g(x_n)\}$. They commute and $f_*$ is invertible. If Brower's fixed point theorem were true for $S_f$ the result would follow. Indeed, a fixed point of $(f_*)^{-1}\circ g_*$ gives the desired orbit.

Solenoid $S_f$ is a non-empty closed subset of $B^{\mathbb Z}$ eqiupped with the product topology. Hence $S_f$ is compact. In fact, $S_f\subset K^{\mathbb Z}$, where $K=\cap_{n>0}f^n(B)$. And I think it is plausible (can anybody give a proof?) that $S_f$ contracts to the orbit of the fixed point of $f$. (later: I am not that sure now)

Unfortunately, Brower's fixed point theorem doesn't hold for compact contractible spaces. But this seem to be quite old and active research area as I learned from the introduction to this paper http://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02071-1/S0002-9947-99-02071-1.pdf People prove positive results under various additional assumptions.

So, maybe it's useful to look at the problem from this point of view.